Tecl^nical   Drawing  Series 


Essentials  of  gearing 


ANTHONY 


<    ooG  Looooo  '^r-'^    ^ 


S-^AT    T.:/,Cf  E3  s  c   L   :<:e 
SA..TA  BARBARA.  CALIFORNIA 
,^^9-?-^ 


Technical  Drawing  Series 


ANTHONY'S  MECHANICAL  DRAWING 

ANTHONY'S    MACHINE  DRAWING 

ANTHONY'S   GEARING 

ANTHONY   AND   ASHLEY'S   DESCRIPTIVE  GEOMETRY 

DANIELS'S   FREEHAND   LETTERING 

DANIELSS  TOPOGRAPHICAL   DRAWING 


D.  C   HEATH   &  CO.,  PUBLISHERS 


TECHNICAL    DRAWING    SERIES 


THE    ESSENTIALS    OF    GEARING 

A  TEXT  BOOK  FOR  TECHNICAL  STUDENTS  AND  FOR  SELF-INSTRUCTION, 

CONTAINING   NUMEROUS   PROBLEMS  AND 

PRACTICAL  FORMULAS 


GARDNER  C.  ANTHONY,  A.M.,  Sc.D. 

Professor  of  Drawing  in  Tufts  College  and  Dean  of  the  School  of  Engineering; 

Author  of  "Elements  of  Mechanical  Drawing,"  and  "Machine  Drawing;" 

Member  of  American  Society  for  the  Promotion  of  Engineering  Education; 

Member  of  American  Society  of  Mechanical  Engineers 


REVISED 


D.    C.    HEATH   &   CO.,   PUBLISHERS 

BOSTON        NEW   YOUK        ClUCAGO 


Copyright, 

Bt  Gardner  C.  Anthony, 

1897  AND  1311. 

2  lO 


A  5  "''  lU^.1-0 

Cjll  PEEFAOE. 


The  most  feasible  method  for  the  acquirement  of  a  working  knowledge  of  the  theor}-  of 
gear-teeth  curves  is  by  a  graphic  solution  of  problems  relating  thereto.  But  it  requires  much 
time  on  the  part  of  an  instructor,  and  is  very  difficult  for  the  student,  to  devise  suitable  exam- 
ples which,  while  fully  illustrating  the  theory',  shall  involve  the  minimum  amount  of  drawing. 
It  is  the  aim  of  the  author  to  overcome  these  difficulties  by  the  presentation  of  a  series  of  pro- 
gressive problems,  designed  to  illustrate  the  principles  set  forth  in  the  text,  and  also  to  encour- 
age a  thorough  investigation  of  the  subject  by  suggesting  lines  of  thought  and  study  beyond 
the  limits  of  this  work. 

In  this  as  in  the  other  books  of  the  series  the  author  would  emphasize  the  fact  tliat  the 
plates  are  not  intended  for  copies,  but  as  illustrations.  A  definite  la^'-out  for  each  problem  is 
given,  and  the  conditions  for  the  same  are  clearly  stated.  This  is  accompanied  by  numerous 
references  to  the  text,  so  that  a  careful  study  of  the  subject  is  necessitated  before  performing 
the  problems. 

Although  specially  addressed  to  students  having  no  previous  knowledge  of  the  principles 
of  kinematics,  it  is  also  designed  to  serve  as  supplementary  to  treatises  on  this  subject. 


iV  PREFACE. 


The  methods  and  problems  have  already  proved  their  usefulness  in  the  instruction  of  stu- 
dents of  many  grades ;  and  it  is  hoped  that  their  publication  may  promote  a  wider  interest  in, 

and  more  thorouerh  study  of,  the  essentials  of  gearing. 

GARDNER   C.  ANTHONY. 
Tufts  College,  Sept.  24,  1897. 


PREFACE   TO   THE   REVISED    EDITION. 

In  revising  The  Essentials  of  Geaeing  it  seemed  desirable  to  supplement  the  work  by 

an  introduction  which  should  better  adapt  it  to  tlie  use  of  college  students.     In  doing  this  the 

chief  aim  of  the  author  has  been  to  present  the  general  principles  underlying  the  study  of 

velocity  ratio,  which  should  serve  not  only  as  an  introduction  to  the  study  of  Gearing,  but  as 

supplementary  to  the  many  excellent  treatises  on  Kinematics  and  Mechanism.     It  is  in  this 

manner  tliat  the  author  makes  use  of  the  book  in  his  classes. 

GARDNER   C.  ANTHONY. 
Tufts  College,  Jan.  2,  1911. 


co:n'te:nts. 


Introduction i^ 

I.  Subject  of  Introduction.  II.  Velocity  Ratio.  III.  Angular  Velocity.  IV.  Instantaneous  Center  or 
Centro.  V.  Determination  and  Proof  of  Instantaneous  Center.  VI.  Centro  of  Rolling  Curve. 
VII.  Velocity  Ratio  as  Deterniined  hy  Intermediate  Connectors.  VIII.  Velocity  Ratio  in  Contact 
Motion.  IX.  Conditionsof  Constant  Velocity  Ratio.  X.  Flexible  Connector.s.  XI.  Similarity 
between  the  Three  Modes  of  Transmission.  XII.  Sliding  Action.  XIII.  Directional  Rotation. 
XIV.    Positive  Rotation.     XV.   Pure  Rolling.     XVI.    The  Three  Modes  of  Transmission. 


CHAPTER   I. 

General  Principles 

I.  Constant  Velocity  Ratio.     2.  Positive  Rotation.     3.  Gearing. 


Odontoidal  Curves 

4.    Classes   of   Curves 


CHAPTER   II. 


5.  Cycloid.  6.  Epitycloid.  7.  Hypocyclold.  8.  To  Construct  a  Normal. 
9.  A  Second  Method  for  Describing  the  Cycloidal  Curves.  10.  Double  Generation  of  the 
Epicycloid  and  Ilypocycloid.      11.    Epitrochoid.      12.    Involute. 


CHAPTER   111. 


Spur  Gears  and  the  Cycloidal  System 


13.    Theory  of   Cycloidal    Action.       14.    Law   of   Tooth    Contact.       15.    Application.       16.    Spur  (Jears. 
17.    Circular  Pitcli.      18.    Diainotcr  Pitch.     19.    Face  or  Addeudinn.     20.    Flank  or  Dedendum. 

V 


VI  CONTENTS. 

PAGE 

21.  Path  of  Contact.  22.  Arc  of  Contact.  23.  Arcs  of  Approach  and  Recess.  24.  Angle  of 
Obliquity  or  Pressure.  25.  Rack.  26.  Spur  Gears  Having  Action  on  Both  Sides  of  the  Pitch 
Point.  27.  Clearance.  28.  Curve  of  Least  Clearance.  29.  Backlash.  30.  Conditions  Govern- 
ing the  Practical  Case.  31.  Proportions  of  Standard  Tooth.  32.  Influence  of  the  Diameter  of 
the  Rolling  Circle  on  the  Shape  and  Efficiency  of  Gear  Teeth.  33.  Interchangeable  Gears. 
34.  Practical  Case  of  Cycloidal  Gearing.  35.  Face  of  Gear.  36.  Comparison  of  Gears,  illus- 
trated in  Plates  4,  5,  and  6.     37.   Conventional  Representation  of  Sj^ur  Gears. 


CHAPTER  IV. 

Involute  System 26 

38.  Theory  of  Involute  Action.  39.  Character  of  the  Curve.  40.  Involute  Limiting  Case.  41.  Epi-- 
cycloidal  Extension  of  Involute  Teeth.  42.  Involute  Practical  Case.  43.  Interference. 
44.  Influence  of  the  Angle  of  Pressure.  45.  Method  for  Determining  the  Least  Angle  of 
Pressure  for  a  Given  Number  of  Teeth  Having  no  Interference.  46.  Defects  of  a  System 
of  Involute  Gearing.      47.    Unsymmetrical  Teeth. 


CHAPTER  V. 

Annular  Gearing 38 

48.  Cycloidal  System  of  Annular  Gearing.  49.  Limiting  Case.  50.  Secondary  Action  in  Annular 
Gearing.  51.  Limitations  of  the  Intermediate  Describing  Curve.  52.  Limitations  of  Exterior 
and  Interior  Describing  Curves.  53.  The  Limiting  Values  of  the  I^xterior,  Interior,  and  Inter- 
mediate Describing  Circles  for  Secondary  Action.  54.  Practical  Case.  55.  Summary  of 
Limitations  and  Practical  Considerations.      56.    Involute  System  of  Annular  Gearing. 


CONTENTS.  Vii 

CHAPTER   YL                                                                           P^vGE 
Bevel  Gearing 45 

57.  Tlieory  of  Bevel  Gearing.  58.  Character  of  Curves  Employed  in  Bevel  Gearing.  59.  Tredgold 
Approximation.  60.  Drafting  the  Bevel  Gear.  61.  Figiiring  the  Bevel  Gear  with  Axes  at  90°. 
62.    Bevel  Gear  Table  for  Shafts  at  90°.     63.    Bevel  Gears  with  Axes  at  Any  Angle. 

CHAPTER  vn. 

Special  Forms  of  Gears,  Notation,  Formulas,  etc 57 

64.  Odontographs  and  Odontograph  Tables.  65.  The  Three-Point  Odontograph.  66.  The  Grant 
Involute  Odontograph.  67.  AVillis's  Odontograph.  68.  The  Robinson  Odontograph.  69.  TJie 
Klein  Coordinate  Odontograph.  70.  Special  Forms  of  Odontoids,  and  Tiieir  Lines  of  Action. 
71.    Conjugate  Curves.     72.    Worm  Gearing.     73.    Literature.     74.    Notation  and  Formulas. 

CHAPTER   YITL 

Problems 70 

75.   Method  to  be  Observed  in  Performing  the  Pioblems  — 

PROBLK.-\r  1.    Cycloidal  f.imiting  Case.     Face  or  Flank  Only 71 

2.  Cycloidal  Limiting  Case.     Face  and  Flank 7^3 

3.  Cycloidal  Gear.     Practical  Case 74 

4.  Involute  Limiting  Case 70 

5.  Involute  Practical  Cases 77 

6.  Cycloidal  Annular  Gear 7s 

7.  Involute  Annidar  Gear 79 

8.  Cycloidal  and  Involute  Bevel  Gears.     Sliafts  at  90° 80 

9.  Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  Other  than  90°       .         .         .         .81 


THE   ESSENTIALS  OF   GEARING. 


INTRODUCTION. 

I.  This  treatise  deals  with  the  principles  governing  the  transmission  of  motion  by  gear- 
ing. The  introduction  treats  of  the  fundamental  principles  governing  constrained  motion  in 
one  plane  as  it  relates  to  the  direction  and  relative  velocities  in  link  and  contact  mechanism, 
and  with  special  reference  to  gearing. 

Those  desiring  a  less  mathematical  treatment  of  the  principles  should  omit  the  introduc- 
tion, as  the  subsequent  consideration  of  the  subject  is  not  necessarily  dependent  thereon. 

The  diagrams  are  lettered  to  correspond  throughout  and,  in  general,  to  accord  with  the 
plates  on  gearing. 

II.  Velocity  Ratio.  In  discussing  the  motion  of  the  parts  of  a  machine  we  are  chiefly  con- 
cerned with  their  relative  velocities,  although  the  path  described  by  a  point,  or  points,  may 
in  some  cases  be  of  equal  or  greater  importance. 

As  the  path  of  any  point  may  be  regarded  as  circular  for  any  instant  of  time,  we  may  com- 
pare the  velocities  of  connected  parts  by  their  respective  revolutions,  linear  velocities  of 
peripheries,  or  the  angles  described  in  a  unit  of  time.  The  I'atio  between  two  such  velocities 
is  called  the  velocity  ratio,  and  the  study  of  gearing  relates  chiefly  to  tlie  methods  used  for 
maintaining  a  constant  ratio  of  velocities  between  two  shafts  while  transmitting  a  positive 
rotation. 


IXTRODUCTION", 


III.     Angular  Velocity  is  the  term  used  to  exjiress  the  velocity  of  a  rotating  bod}'.     It  may 
be  expressed  as  follows  : 

1.  Number  of  revolutions  per  unit  of  time;   as  120  rev.  per  n 

2.  Number  of  degrees  per  unit  of  time  ;  as  720°  per  sec. 

3.  Number  of  radians  per  unit  of  time  ;  as  12.56  radians  per  sec. 
These  values  are  the  same  for  all  points  in  a  revolving  body. 
The  radian  is  a  unit  for  angular  measure.      It  is  an  arc  of  linear 

measure  equal  to  the  radius,  and  subtends  57.3°, 

hence  there  are  2  tt  radians  in  360°.     A  rotating 

body  making  N  revolutions  per  unit  of  time  has 

an  angular  velocity  (co)  of  2  7rN  radians.      This 

,    ,     linear  velocitv      'IrrrH      v      v' 
IS  equal  to 


=  -  =  -  =  «, 


LINEAR  VELOCITY  OF  A 

Fig:.  I. 


radius  r 

V  and  I' '  being  the  linear  velocities  of  points  at  ;•  and  r'  radial  distance  from  the  center  of  rotation. 

Since  co  is  the  same  for  all  values  of  r.  it  is  equal  to  -  and,  therefore,  numerically  equal  to  the 

linear  velocity  of  a  point  at  unit  distance  from  the  center. 

If  the  scale  of  velocities  in  Fig.  I  be  100  ft.  per  minute  per  inch,  the  number  of  radians 

will  be  -  =  ^^  =  2G6.6.  and  the  number  of  revolutions  per  minute  would  be  —  =  "         =  42.5. 
r      .<o  '■  2-77       6.28 

Example.     A  disk  60"  diameter  makes  300  revolutions  per  minute,      (a)  What  is  its 

angular  velocity  ?     (J)   What  is  the  linear  velocity  of  a  point  in  its  circumference  ? 

SoLUTiox.      (a)  tfj  =  2  ttN  =  6.28  x  300  =  188-4  radians  per  minute.     (6)  i'  =  &)r=1884  x  2| 

=  4710  ft.  per  minute. 


IXTKODUCTIOX. 


XI 


IV.  Instantaneous  Center,  or  Centro.  All  bodies  in  motion  may  be  considered  as  rotating 
about  some  axis  which  can  be  determined  for  any  instant  of  time.  This  axis  is  called  the 
instantaneous  axis,  or  axode,  and,  being  known,  enables  one  to  determine  the  direction  of 
motion  and  velocity  of  any  point  in  a  body,  since  the  angular  velocity  is  constant  for  all 
points  in  the  same  body.  If  tlie  motion  considered  is  in  a  plane,  the  rotation  takes  place 
about  a  point  known  as  the  instantaneous  center,  or  centro. 

EK,  Fig.  II,  is  a  line  having  motion  in  the  })lane 
of  the  paper,  point  E  moving  in  the  direction  ER  with  /'\ 

a  velocity  equal  to  ER,  and  point  K  moving  in  the        ^   z^-'"'  \ '       

direction  KL  with  a  velocity  to  be  determined.     Point    -{-— ^- — ^^'-\ ^"^ ' 

E  may  be  considered  as  rotating  about  some  point  in        l\    j^^  J>^      I 

EO    which    is    perpendicular    to    ER   at    E,   since    the 
direction  of  motion  of  any  point  in  a  rotating  body 

is  perpendicular  to  the  radius  at  tliat  point.     Simi-  /^^T  ^ ^ 

larly   K   may  be  considered  as   rotating  about  any  ^^' 

point  in  KO  perpendicular  to  KL  at  K.  Their  point  of  intersection  0  will  be  common  to  both 
radii,  and  therefore  the  center  of  rotation  of  EK  for  the  instant ;  hence,  the  instantaneous 
center,  or  centro.      According  to  Art.  Ill,  all  points  in  the  line  will  have  an  angular  velocity 


^  V        ER        KL      , 

equal  to  -  =  —  =  —  ;   lience    KL 
^  r      OE      OK 


ER 

OE 


X  KO.      KL  may  be  graphically  obtained  by  laying  off  OE' 


on  OK,  equal  to  OE,  and  drawing  E'R'  perpendicular  to  OK  and  equal  to  ER.      The  intersection 
of  the  radial  OR'  and  KL  will  determine  the  velocity  of  point  K,  which  is  KL. 

Again,  if  we  determine  EH,  the  longitudinal  component  of  ER,  it  must  equal  KM,  the  longitu- 
dinal component  of  KL,  otherwise  the  line  EK  would  not  maintain  a  constant  length. 


Xll 


INTRODUCTION. 


V.     Determination  and  Proof  of  the  Instantaneous  Center.     Fig.  III.     EK  is  an 
and  inexteusiblc  rod.     KL  indicates  the  motion  of  point  K  in  direction  and  intensity. 
ER  indicates  the  motion  of  point  E,  the  longitudinal  components  of  l 

ER  and  KL,  which  are  EH  and  KM,  being  eqnal. 

Draw  IT,  and  Z  will  be  the  point  about  wliich  the  rod  tends  to 
rotate  by  reason  of  the   side   components   Kl   and   ET.     YZ  is  the 
horizontal  component  of  point  Z  and,  in  common  Avith  all  points 
in  the  rod,  is  equal  to  EH  and  KM.     B'rom  K  draw  KG  perpendicular 
to  KL,  and  from  Z  draw  ZO  perpen- 
dicular to  YZ.     If  the  angle  LOK 
equals  the  angle  YOZ,  points  Z  and 
K    will    have    the    same    angular 
velocity  about  0  (Art.  Ill),  and 
hence  0  will  be  the  instantaneous 
center. 


inflexible 

Similarly 


=  — ,  and 

oz 


Proof.     In  the  similar  triangles  LKl  and  KOZ,  —  =  — ,  but  IL  =  MK  =  YZ,  hence 
therefore  angle  LOK  is  equal  to  angle  YOZ.     q.e.d. 

Similarly  angle  ROE  =  angle  YOZ  =  angle  LOK,  hence  0  must  be  the  center  about  which  all 
points  in  the  rod  EK  revolve  with  equal  angular  velocity. 


VI.    If  a  circle,  or  other  curve,  rolls  on  a  right  line,  or  curve,  its  centre  at  any  instant 
will  be  the  point  of  tangency  of  the  rolling  and  fixed  curve,  since  this  will  be  the  only  point 


INTRODUCTION. 


Xlll 


of  the  circle  whicli  is  at  rest.      In  Fig.  IV  the  rolling  circle  MAO  has  its  centro  at  0,  its  point 
of  tangency  with  the  line  on  which  it  rolls.     This  may  be  determined  by  finding  the  direc- 
tion of  motion  of  any  two  points  in  the 
curve,  such  as  A  and  R,  and  drawing  per- 
pendiculars   from    these    points.       Their 
intersection    will    be    the    instantaneous 
center  of  rotation,  or  centro.     If  CD  be  the 
velocity  of  translation,  it  will  also  be  the 
linear  velocity  of  all  points  in  the  circum- 
their  rotation   about    the    center   C.     Point  A 
will,    therefore,  have    components    of  velocity  AF   and   AE 
equal    to    CD,  and    point  R  will  have  components  RT  and 
RV  also  e(|ual    to    CD.      Perpendiculars    to    the    resultants 
AB  and  RS  from  points  A  and   R   will    intersect    at    0,  the 
centro. 

Again,  if  points  C  and  M  be  taken,  the  centro  will  be 
determined  as  follows :  since  both  points  move  in  horizontals,  the  point  of  intersection  of  the 
perpendiculars  from  tliese  points  will  be  indeterminate,  and  other  means  must  be  emploj-ed 
to  determine  the  centro.  The  velocities  of  points  C  and  M  will  be  CD  and  MN.  The  latter  is 
the  resultant  of  two  liorizontal  components  equal  to  CD,  one  being  the  circumferential  velocity, 
and  the  otlier  the  velocity  of   translation  common  to  all   points  of   the  circle.      Therefore, 

MN  =  2CD;   but        =       ,  hence  0  must  be  the  point  about  which  rotation  takes  place. 
CD      CO 


Fig-.  IV. 


XIV 


INTRODUCTION". 


VII.  Velocity  Ratio  as  Determined  by  Intermediate  Connectors.  F  and  G,  Fig.  V,  are  the 
fixed  centers  of  the  revolving  arms  1  and  2,  their  free  ends  being  connected  by  the  link  EK. 
It  is  required  to  determine  the  velocity  ratio  between  1  and  2  for  the  instant  shown. 

Peoof.  Let  ER  be  the  graphic  rejDresentation  of  the  linear  velocity  of  the  driving  arm  1, 
the  longitudinal  component  of  which  is  EH.     Since  the  length  of  the  link  is  constant,  the 


Fig.  V. 


longitudinal  component  of  the  motion  of  K  in  arm  2  must  be  KM  =  EH,  and  the  linear  velocity 


ER 
will  be  KL.      Let  co^  and  co.^  designate  the  angular  velocities  of  arms  1  and  2,  then  Wj  =  —  and 

cOr,=  — .      From  tlie  similar  triansrles  EHR  and  FCE,  — =  —     Also  from  the  similar  triangfles 
-       GK  ^  EF       FC  ^ 

1     ^.,„      KL       KM       EH      1  CO,       ER       GK       EH       GN       GN  ^^         f  .^  •     -i 

KML    and    GNK,    —  =  —  =  — ,  hence    -^  =  — X —  =  — x —  =  —       Also    from    the    similar 
GK       GN       GN  &)o       FF       KL       FC       EH        FC 


INTRODUCTION. 


XV 


triangles  AFC  and  AGN,        =  —     Hence  the  general  law:    The  angular  velocities  of  revolving 

arms  with  an  intermediate  connector  are  inversely  proportional  to  the  segments  into  tvhich  the  line 
of  action  divides  the  line  of  centers. 

This  law  may  be  obtained  also  by  using  the  centro  0  in  connection  with  centers  F  and  G. 
Let  ©3  designate  the  angular  velocity  of  E  and  K  about  the  instantaneous  center  0.  Di'aw  OZ 
perpendicular  to  the  line  of  the  link  EK  and  observe  the  similarity  of  triangles  EHR  and  OZE, 


also   of    KML   and    OZK. 


ER 
EF 


KL 
GK 


ER        KL 

(Wo  =  —  = 

OE       OK 


"3 


ER       OE 

-  X  — 

EF        ER 


^  — 


KL       OK 


^3 


OK_OZ 
<»3       GK       KL       GK~GN 

If  the  center  line  of  the  link 
intersects  the  line  of  centers 
between  the  fixed  centers,  as 
in  Fig.  VI,  the  only  change  in 
the  conditions  will  be  in  tlie 
direction  of  rotation  of  arms  1 
and  2.  In  Fig,  V  they  are 
alike  and  in  Fig.  VI  they  are 
opposite.  As  tlie  same  letter- 
ing is  used  in  both  figures,  the 
demonstration  may  be  applied 
to  either. 


<y,        ft),       0).        OZ       GN        GN 

hence  -^x— ^=— !^=  —  x  —  = 


which  is  equal  to 


OE 
EF  ' 
AG 


OZ 
FC 


Also 


XVI 


INTRODUCTION. 


VIII.  Velocity  Ratio  in  Contact  Motion.  Fig.  VII.  F  and  G  are  the  fixed  centers  of 
revolving  arms  1  and  2  having  curved  surfaces  in  contact.  Motion  is  imparted  from  1  to  2 
by  contact  between  the  curved  ends,  the  direction  of  rotation  being  indicated  by  arrows. 
It  is  required  to  determine  the  velocity  ratio  at  the  instant  shown,  and  to  deduce  tlie  general 
law. 


Fi&.  VII. 


At  the  point  of  contact  B  draw  the  common  tangent  BT,  and  the  normal  BW,     Let  Bl,  per- 
pendicular to  BF  at  B,  represent  the  linear  velocity  of  point  B  in  arm  1,  the  angular  velocity 


being 


Bl_ 

bf" 


Resolve  Bl  into  the  tangential  component  BV  and  normal  component  BW.     The 


direction  of  motion  of  point  B  in  arm  2  will  be  perpendicular  to  BG,  and  the  normal 
component  BW  will  be  the  same  as  for  point  B  in  arm  1.  If  this  were  not  so,  arm  1  would 
recede  from,  or  compress,  arm  2.     The  tangential  component  must  lie  in  BT  and  the  value 

BS 

of  the  resultant  motion  will  be  BS,  the  angular  velocity  of  arm  2  being 

BG 


The  ratio  between  the  angular  velocities  of  arms  1  and  2  will  be  -^  =  —  x  — 

&j.,       BF       BS 


From  the 


INTRODUCTION. 


XVU 


BW 


BG       GN 


similar  triangles  IBW  and  BFC,  —='f^;  also  from  the  similar  triansrles  SBW  and  BGN,  —  = 

BF       FC  ^  BS        BW 


Substituting  in  the  above, 


BW      GN        GN    ,     ^  ^  ^1^-1  1  GN       AG 

—  X  —  =  — ,  but  from  the  triangles  AGN  and  AFC,  —  =  -^ 
PC       BW       FC  ^  FC       AF 


Hence  the  general  law:    The  angular  velocities  of  revolving  arms  in  contact  motion  are  in- 
versely proportional  to  the  segments  into  which  the  common  normal  divides  the  line  of  centers. 

In  Fig.  VII  the  contact  is  such 
that  the  arms  rotate  in  the  same 
direction,  but  if  the  rotation  be 
opposite,  the  conditions  may  be 
similar  to  Fig.  VIII,  which  is  let- 
tered to  correspond  with  Fig. 
VII,  and  the  same  demonstration 
may  be  employed. 

IX.     Condition  of  Constant  Velocity  Ratio.     From   V^ 
the  above  law  it  will  be  seen  that  any  change  of  point  | 

A  will  change  the  velocity  ratio,  hence  :  To  preserve  a  constant  velocity  ratio  the  common  normal^ 
or  line  of  action^  must  always  cut  the  line  of  centers  in  the  same  point. 


Fig.  VIII. 


X.  Flexible  Connectors.  If  pulleys  be  substituted  for  the  revolving  arms  of  the  link 
mechanism  in  Fig.  V,  and  if  a  flexible  connector  such  as  a  belt,  rope,  or  chain  be  used  to  re- 
place the  link,  the  velocity  ratio  may  be  maintained  constant.     And  if  the  ratio  of  the  pulley 

diameters  be  that  of  — -  or  — ,  the  velocity  ratio  will  be  the  same  as  that  of  the  link  mecha- 
FC         AF 


XYUl 


INTRODUCTION. 


nism.  Figure  IX  illustrates  this  case.  FG  is  the  fixed  link,  FC  and  GN  (the  radii  at  the  point  or 
tangency  with  the  flexible  link)  are  the  rotating  arras,  and  CN  the  connector  link.  If  1  be  the 
driver,  the  driving  side  of  the  belt,  or  flexible  connector,  will  be  CN',  but  in  other  respects 
the  action  of  this  mechanism,  for  the  instant,  does  not  differ  from  that  of  Fig.  V  and  Fig.  VII. 
Figure  X  illustrates  a  case  analogous  to  that  represented  by  Fig.  VI  and  Fig.  VIII 

C 

STRIVING  ^ 


Tig.  IX. 


XI.  Similarity  between  the  Three  Modes  of  Transmission.  Figure  XI  illustrates  a  combina- 
tion of  Figs.  V,  VII,  and  IX.  It  will  be  observed  that  the  center  line  of  link  EK,  the  normal 
component  BW,  and  the  line  of  the  belt  CN  all  coincide,  and  that  the  common  normal,  or  line 
of  action,  intersects  the  line  of  centers  AG  in  the  point  A,  thus  determining  the  common  ve- 

locity  ratio,  — '  =  — 
&)„      AF 


Only  one  of  these,  the  pulleys,  will  surely  maintain  a  constant  velocity 


ratio,  although  the  direct  contact  mechanism  may  be  made  to  do  this  by  so  shaping  the 
contact  curves  that  their  common  normals  will  continue  to  pass  tlirough  point  A. 


INTRODUCTION. 


XIX 


■Fig.  XI. 


XX  INTRODUCTION. 

XII.  Sliding  Action.  Figures  XII,  XIII,  and  XIV  illustrate  the  sliding  action  taking  place 
between  surfaces  of  contact.     They  are  lettered  to  correspond  with  the  preceding,  and  in 

each  case  the  velocity  ratio  of  driver  to  follower  is  _.     The  velocity  of  the  drivers  at  the  point 

AF 
of  contact  is  Bl  and  of  the  followers,  BS ;  the  common  normal  component  is  BW,  and  the  tan- 
gential components  of  driver  and  follower,  on  which  the  sliding  action  depends,  are  BV  and  BT. 
It  will  be  observed  that  in  Fig.  XII  the  tangential  components,  BV  and  BT,  act  in  opposite 
directions,  and  the  amount  of  sliding  action  will  equal  BV  +  BT.  In  Fig.  XIII  they  act  in  the 
same  direction,  and  the  total  action  is  BV  —  BT.  In  Fig.  XIV  they  are  equal,  and  alike  in  di- 
rection, and  therefore  no  sliding  action  takes  place.  If  rotation  in  one  direction  be  considered 
positive,  and  the  opposite  direction  negative,  the  sliding  action  is  equal  to  the  algebraic  difference 
of  the  tangential  components. 

XIII.  Directional  Rotation.  The  relative  direction  of  rotation  is  easily  determined  by 
observation,  but  since  all  phases  of  the  action  can  best  be  referred  to  the  line  of  action,  the 
following  law  should  be  observed  :  The  directional  rotation  of  driver  and  follower  are  alike  when 
the  fixed  centers  lie  on  the  same  side  of  the  line  of  action^  and  opposite  if  the  centers  lie  on  opposite 
sides  of  the  line  of  action. 

XIV.  Positive  Rotation.  If  one  cylinder  transmits  motion  to  a  second  by  contact,  we 
know  that  motion  is  not  positive,  since  it  is  dependent  on  the  friction  due  to  the  imper- 
fection of  the  contact  surfaces.  In  the  cases  illustrated  by  Figs.  XII,  XIII,  and  XIV,  a 
positive  rotation  is  apparent  in  the  positions  shown,  but  by  continuing  the  rotation,  a  point 
will  be  reached  at  which  the  driver  will  cease  to  move  the  follower.     This  will  occur  only 


INTRODUCTION. 


ZXl 


when  the  driving  contact  point  has  no  com- 
ponent in  the  direction  of  the  normal,  or  line 
of  action.  In  this  case  Bl  will  coincide  with 
the  tangent  at  the  point  of  contact,  and  the 
direction  of  the  normal  will  be  through  the 
center  F.  The  dotted  position  of  the  contact 
mechanism  in  Fig.  XIII  represents  this  phase, 
and  the  normal  B'F  passes  through  the  center 
of  the  follower,  since  the  normal  component  is 
always  perpendicular  to  the  common  tangent 
of  the  contact  surfaces.  Whenever  this  con- 
dition occurs,  positive  driving  will  cease  and 
the  law  may  be  expressed  as  follows: 

Positive  rotation  can  take  place  only  when 
the  common  normal^  or  line  of  action^  does  notjMSS 
throiKjh  the  fixed  center  of  driver,  or  follower. 

XV.  Pure  Rolling.  From  the  considera- 
tion of  Fig.  XIV,  in  Art.  XII,  it  was  observed 
that  there  was  no  sliding  action  when  the  tan- 
gential components  were  alike  in  magnitude 
and  direction.     Tliis  can  take  place  only  when 


Fig.  XIV. 


XXU  INTRODUCTION 

the  contact  is  on  the  line  of  centers.  If  such  contact  is  maintained,  the  result  will  be  pure 
rolling  such  as  exists  between  two  cylinders,  or  circles. 

Figures  XV,  XVI,  XVII  are  examples  of  pure  rolling.  Fig.  XV  illustrates  a  pair  of 
logarithmic  spirals  rotating  about  centers  F  and  G,  with  contact  at  B.  The  character  of  these 
curves  is  such  that  the  angle  between  the  tangent  and  radius  at  any  point  is  a  constant ;  the 
locus  of  the  points  of  contact  will  be  FG,  and  there  will  be  pure  rolling  with  positive  rotation, 
but  not  a  constant  angular  velocity  ratio.  As  these  curves  are  not  closed,  they  cannot  be 
used  to  transmit  continuous  motion. 

Figure  XVI  illustrates  two  elliptical  curves  revolving  about  their  foci.  The  contact  will 
always  be  on  the  line  of  centers  FG,  and  therefore  pure  rolling  will  take  place.  Tlie  angular 
velocity  ratio  will  not  be  constant,  but  there  will  be  positive  rotation  save  when  the  major 
axes  coincide  with  the  line  of  centers,  at  which  time  the  common  normal  will  pass  through 
the  fixed  centers.     Art.  XIV. 

Figure  XVII  differs  from  the  preceding  in  that  pure  rolling  is  accompanied  with  a  constant 
velocity  ratio,  but  we  have  lost  the  positive  rotation. 

XVI.  The  Three  Modes  of  Transmission.  Constant  velocity  ratio,  positive  rotation,  and 
pure  rolling  cannot  be  maintained  at  the  same  time  in  any  one  mechanism.  Only  two  of  the 
three  conditions  are  possible.  Figs.  XV,  XVII,  XVIII  illustrate  these  combinations.  In 
Fig.  XV  there  is  positive  rotation  and  pure  rolling,  in  Fig.  XVII  there  is  constant  velocity 
ratio  and  pure  rolling,  and  in  Fig.  XVIII  there  is  a  constant  velocity  ratio  and  positive 
rotation.      The  last  condition  is  the  one  to  be  maintained  in  the  design  of  gear  teeth. 


INTRODUCTION. 


XXlll 


Fig:.  XVIII. 


Fig.  XVII. 


THE   ESSENTIALS   OF   GEARING. 


CHAPTER   I. 

GENERAL    PRINCIPLES. 

1.  Constant  Velocity  Ratio.  Motion  may  be  transmitted  between  lines  of  shafting  by 
means  of  friction  surfaces  ;  and  if  there  be  no  slipping  of  the  contact  surfaces,  the  circumference 
of  the  one  will  have  the  same  velocity  as  the  circumference  of  the  other.  The  number  of  revo- 
lutions of  the  shafts  will  be  invei-sely  proportional  to  the  diameter  of  the  friction  surfaces,  and 
this  ratio  will  l)e  maintained  constant  under  the  condition  of  no  slip.  Such  friction  surfaces 
and  shafts  are  said  to  have  a  constant  velocity  ratio. 

2.  Positive  Rotation.  In  order  to  transmit  force,  as  well  as  motion,  and  to  insure  its 
being  positive,  it  will  be  necessary  to  place  cogs,  or  elevations,  on  one  of  the  friction  sur- 
faces, and  make  suital»le  depressions  in  the  other  surface. 

3.  Gearing.  The  study  of  toothed  gearing  is  a  study  of  the  shape  of  these  cogs,  teeth,  or 
odontoids,  which  are  designed  to  produce  a  positive  rotation  while  preserving  the  condition  of 
constant  velocity  ratio. 

1 


GEARS    CLASSIFIED. 


Fig.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


Gears  may  he  classified,  as  fol- 
lows :  — 

1.  If  tlie  shafts  are  parallel, 
the  friction  surfaces  would  be 
cylinders  (Fig.  1),  and  the  gears 
designed  to  produce  the  same 
condition,  as  to  the  velocity,  are 
called  Sjjur   Grears  (Fig.  2). 

2.  If  the  shafts  intersect,  the 
friction  surfaces  would  be  cones 
(Fig.  3),  and  the  gears  called 
Bevel   Grears  (Fig.  4). 

3.  If  the  shafts  are  .neither  in- 
tersecting nor  parallel,  the  friction 
surfaces  will  be  hyperboloids  of 
revolution  (Fig.  5),  and  the  gears 
called  Hyperlolic,  or  Skew  Crears 
(Fig.  6). 

In  the  preceding  cases  the  ele- 
ments of  the  teeth  are  rectilinear, 
and  the  friction  surfaces  touch 
each  other  along  right  lines. 

4.  If  the  elements  of  the  teeth 


GEARS    CLASSIFIED. 


in  eitlier  of  tlie  first  three  cases 
be  made  lielical,  an  entirely  dif- 
ferent class  of  gearing  will  result. 
The  various  forms  are  known  as 
Twisted^  /Spiral,  Worm,  and  Screw 
G-earing  (Figs.  7  and  8).  The 
action  of  the  latter  is  analogous  to 
that  of  a  screw  and  nut. 

One  of  these  forms  is  generally 
employed  as  a  substitute  for  hy- 
perbolic, or  skew  geare,  by  reason 
of  the  difficulty  experienced  in  cor- 
rectly forming  the  teeth  of  such 
gears. 

5.  Another,  although  but  little 
used,  form,  is  tliat  known  as  Face 
G-ear'uKj.  The  teeth  are  usually 
pins  secured  to  the  face  of  cir- 
cular disks  having  axes  perpen- 
dicular. The  action  takes  place 
at  a  point  only. 

None  of  the  latter  forms  can 
be  represented  by  friction  surfaces. 


Fig.  7. 


Fig.  8. 


4  ODONTOIDAL    CURVES. 

CHAPTER   11. 

ODONTOIDAL      CURVES. 

4.  The  two  classes  of  curves  coninionly  employed  in  gear  teeth  are  the  cycloidal  and  the 
involute.  A  knowledge  of  their  characteristics  and  methods  of  generating  is  essential  to  an 
undei-standing  of  their  application  in  gearing. 

5.  Cycloid.  Plate  1,  Fig.  1.  The  cycloid  is  a  curve  generated  by  a  point  in  the 
circumference  of  a  circle  which  rolls  upon  its  tangent.  The  circle  is  called  the  describing, 
or  generating  circle,  and  the  point  is  known  as  the  describing,  or  generating  point.  In 
Fig.  1,  Plate  1,  B  is  the  describing  point,  and  B  D  C  E  the  describing  circle,  which  rolls  on 
its  tangent  E  B'". 

Assume  a  point,  C,  on  the  describing  circle,  and  conceive  the  motion  of  the  circle  to  be 
from  left  to  right.  As  it  rolls  upon  its  tangent,  the  arc  E  C  will  be  measured  off  on  E  B'" 
until  point  C  becomes  a  point  of  tangency  at  C.  The  center  of  the  describing  circle  will  now 
lie  at  A',  in  the  perpendicular  to  E  B'"  at  C. 

From  center  A',  with  radius  of  describing  circle,  draw  the  new  position  of  describing  circle. 
The  generating  point  nmst  lie  in  this  circle  at  a  distance  from  C  ec|ual  to  the  chord  B  C, 

Therefore,  with  radius  equal  to  this  chord,  from  center  C,  describe  an  arc  intersecting  the 
new  position  of  the  describing  circle.  The  line  B'  C  is  called  the  instantaneous  radius,  or 
normal,  of  the  curve  at  B',  it  being  a  perpendicular  to  the  tangent  of  the  curve  at  this  point. 


CYCLOIDAL    CURVES.  0 

The  normal  at  B"  would  be  B"  D'.  The  radius  A'  B'  is  known  as  the  describing^  or  generating 
radius^  and  A'  C  is  the  contact  radius,  or  the  radius  at  the  point  of  contact.  In  like  manner 
other  positions  of  the  describing  point  may  be  found,  and  the  curve  connecting  them  will  be 
the  cycloid  required. 

6.  Epicycloid.  Plate  1,  Fig.  2.  If  the  describing  circle  rolls  upon  the  outside  of  an 
arc,  or  circle,  called  the  director  circle,  the  curve  generated  will  be  an  epicycloid,  Fig.  2, 
Plate  1.  The  method  of  describing  this  curve  is  similar  to  that  for  the  cycloid,  and  the 
lettering  is  the  same.  It  must  be  observed,  however,  that  any  contact  radius,  as  A'  C,  is  a 
radial  line  of  the  circle  on  which  it  rolls. 

7.  Hypocycloid.  Plate  1,  Fig.  3.  If  the  describing  circle  rolls  upon  the  inside  of  a  cir- 
cle, the  curve  generated  will  be  an  hypocycloid.  Fig.  3  illustrates  this  curve,  the  same  letter- 
ing being  used  as  that  of  the  preceding  cases. 

If  it  be  required  to  draw  a  normal  at  any  point  of  this,  or  the  two  preceding  curves,  the  fol- 
lowing method  may  be  employed  :  — 

8.  To  Construct  a  Normal.  From  the  given  point  on  the  curve,  as  a  center,  with  radius  of 
generating  circle,  describe  an  arc  cutting  the  path  described  by  the  center  of  the  generating 
circle.  From  this  point  draw  the  contact  radius,  thus  obtaining  the  contact  point.  Connect 
this  with  the  given  point,  and  the  line  will  be  the  required  normal. 

9.  A  Second  Method  for  Describing  the  Cycloidal  Curves.  Plate  2,  Fig.  1.  A  B  c  is  a 
director  circle,  A  D  E,  the  generating  circle  for  the  epicycloid  A  A'  A"  H ,  and  A  K  L  the  generating 
circle  for  the  hypocycloid  A  L  C. 


6  CYCLOIDAL    CURVES. 

To  describe  the  epic3'cloid,  assume  any  point,  D,  on  the  generating  circle,  and  la}-  off  the  arc 
A  D'  on  llie  director  circle,  making  it  equal  to  arc  AD.  If  A  be  the  describing  point,  then  A  D 
will  be  the  normal  when  D  shall  have  become  a  contact  point,  as  at  D'.  With  L  as  a  center,  de- 
scribe the  arc  D  A'.  The  describing  point  A  must  be  in  this  arc  when  D  shall  be  at  D'.  From  D' 
as  a  center,  with  radius  equal  to  the  chord  A  D,  describe  an  arc  intei-secting  A'  D,  and  thus  deter- 
mine A',  a  point  in  the  epicycloid.      Similarly  obtain  other  points,  and  draw  the  curve. 

The  hypocycloid  may  be  constructed  in  like  manner,  as  shown  by  the  same  figure.  This 
also  illustrates  a  special  case  in  A\"hich  the  hypocycloid  is  a  radial  line,  A  L  C,  and  this  is  due 
to  the  diameter  of  the  describing  circle  being  equal  to  the  radius  of  the  director  circle. 

The  same  method  may  also  be  employed  in  the  construction  of  the  cycloid. 

10.  Double  Generation  of  the  Epicycloid  and  Hypocycloid.  Plate  2,  Fig.  1.  The  epi- 
cycloid may  always  be  generated  by  either  of  two  describing  circles,  which  differ  in  diameter 
by  an  amount  equal  to  the  diameter  of  the  director  circle.  Thus  in  the  case  illustrated,  the 
epicycloid  A  A'  A"  H  may  be  generated  by  the  circle  A  D  E,  with  A  as  a  describing  point,  or  by 
the  circle  S  T  H,  with  H  as  a  describing  point.  Similarly  the  hypocycloid  is  capable  of  being 
generated  by  either  of  two  rolling  circles,  the  sum  of  which  diameters  must  equal  that  of  the 
director  circle.* 

11.  Epitrochoid.  Plate  2.  Fig.  2.  When  the  describing  point  does  not  lie  on  the  cir- 
cumference of  the  generating  circle,  a  curve,  commonly  called  an  epitrochoid,  is  described.  If 
the  point  lies  without  the  circle,  as  at  B ,  a  looped  curve,  B  B'  B",  called  the  curtate  epitrochoid, 

*  For  the  geometrical  demonstratiou  of  this  problem  see  the  appeudix  of  Professor  MacCord's  "  Kinematics,"  page  319. 


INVOLUTE    CURVE.  7 

is  described;  and  if  the  point  be  within,  as  at  D,  tlie  curve  will  be  a  prolate  epitrochoid,  as 
D  D'  D". 

To  obtain  a  point  in  the  former,  assume  any  point,  C,  in  the  circumference  of  the  describing 
circle,  and  determine  its  position,  C,  when  it  shall  have  become  a  contact  point.  Draw  the 
contact  radius  A'  C,  and  from  C  and  A' as  centei-s,  with  radii  A  B  and  C  B,  describe  arcs  intei-sect- 
ing-  at  B',  a  point  in  the  curve.  B'  C  is  the  normal  at  this  point.  In  like  manner  obtain  the 
point  D'  in  the  prolate  epitrochoid. 

12.  Involute.  Plate  1,  Fig.  4.  The  involute  is  a  curve  generated  bv  a  point  of  a  tan- 
gent right  line  rolling  upon  a  circle,  known  as  the  base  circle,  or  the  describing  point  may  be 
regarded  as  at  the  extremity  of  a  fine  wire  which  is  unwound  from  a  cylinder  corresponding  to 
the  base  circle.  In  Fig.  4,  A  B  C  D  is  the  arc  of  a  base  circle,  and  A  the  point  from  which  the 
involute  is  generated. 

Layoff  arcs  A  B,  B  C,  C  D,  preferably  equal  to  each  other,  and  from  points  B,  C.  and  D,  draw 
tangents  equal  in  length  to  the  arcs  A  B,  A  C,  and  A  D.  A  line  drawn  through  the  extremity  of 
these  tano-ents  will  be  an  involute  of  the  base  circle  A  B  C  D. 


8  THE    CYCLOIDAL    SYSTEM. 

CHAPTER   III. 

SPUR  GEARS  AND  THE  CYCLOIDAL  SYSTEM. 

13.  Theory  of  Cycloidal  Action.  Plate  3,  Fig.  1.  Let  H  K  and  M  L  be  the  peripheries 
of  two  disks,  having  centers  G  and  F,  and  S  the  center  of  a  third  disk,  also  revolving  in  contact 
with  the  arcs  H  K  and  M  L.  The  largest  disk  will  be  known  as  disk  1,  the  second  size  as  disk 
2,  and  the  smallest  as  disk  3,  or  the  describing  disk.  Consider  the  peripheries  of  these  disks  in 
contact  at  A,  so  that  motion  imparted  to  one  will  produce  an  equal  motion  in  the  circumference 
of  the  other  two,  thus  maintaining  at  all  times  an  equal  circumferential  velocity,  or  constant 
velocity  ratio. 

Imagine  this  to  represent  a  model,  disk  1  having  a  flange  I  0  extending  below  the  other 
disks,  and  the  describing  disk  as  being  provided  with  a  marking  point  at  A,  each  of  the  disks 
being  free  to  revolve  about  their  respective  axes.  Consider  fii"st  the  relation  between  the 
describing  disk  and  disk  1,  the  marking  point  being  at  A.  Suppose  motion  to  be  given  disk  3 
in  the  direction  indicated  by  the  arrow,  so  that  the  describing  point  shall  move  from  A  to  A'. 
The  point  C  of  disk  1,  which  coincides  wdth  A  wlien  the  describing  disk  is  in  the  first  position, 
will  now  have  moved  on  the  circumference  H  K,  to  C,  an  arc  equal  to  A  A'.  During  this  time, 
the  curve  A'  C  will  have  been  drawn  upon  the  flange  of  disk  1  by  the  marking  point.  Next, 
suppose  the  marking  point  to  move  from  A'  to  A",  then,  since  the  circumferences  of  these  disks 
traverse  equal  spaces  in  equal  times,  C  will  have  revolved  to  C",  and  the  curve  A'  C  will  now 
occupy  the  position  E"  C".    But,  since  the  marking  point  has  continued  to  describe  a  curve  upon 


CYCLOIDAL    ACTION.  U 

the  flange  of  disK  1,  the  curve  E"  C"  will  be  extended  to  A".  In  like  manner  the  marking  point 
moves  to  A"',  continuing  to  describe  a  curve,  as  C"  A"  revolves  to  C"  A"'.  If  now  the  describing 
disk  be  freed  from  the  axis  on  which  we  have  supposed  it  to  revolve,  and  be  rolled  on  the  cir- 
cumference H  K,  the  marking  })()int  would  describe  the  same  curve,  A'"  E'"  C",  as  that  already 
drawn,  which  is  an  epicycloid. 

In  the  same  manner,  we  may  imagine  the  marking  point  to  describe  a  curve  upon  disk  2, 
which  curve,  in  its  successive  positions,  would  be  shown  by  A'  B',  A"  B",  and  A'"  B'".  For  the 
same  reason,  too,  the  arc  A  A'  k"  A'"  will  equal  the  arc  B  B'  B"  B'" ;  and  if,  in  a  manner  similar  to 
the  2)i"eceding,  we  roll  the  describing  disk  on  the  inside  of  the  arc  M  L,  we  shall  describe  the 
same  curve  A'"  D"'  B'",  and  find  it  to  be  an  hypocycloid. 

Again,  consider  these  curves,  A'"  C"  and  A'"  B'",  as  being  traced  at  the  same  time  b}-  the 
describing  point  A.  If  we  now  observe  any  special  position  of  the  point,  as  A",  it  will  be  seen 
to  be  common  to  an  epicycloidal,  and  a  hypocycloidal  curve,  which  have  a  common  normal, 
A"  A,  intersecting  the  line  drawn  through  the  centers,  F  and  G,  at  the  point  of  tangency  of  the 
disks.     This  condition  is  true  for  all  positions  of  the  two  curves. 

If  these  curves  A'"  C",  and  A'"  B'",  be  now  used  as  the  outlines  for  gear  teeth,  as  in  Fig.  2,  G 
and  F  being  the  centers  and  H  K  and  M  L  the  pitch  lines,  we  shall  have  obtained  a  positive  rota- 
tion with  a  uniform  velocity  ratio,  for  it  was  under  this  latter  condition  that  the  curves  were 
generated,  and  the  common  normal  to  tlie  curves  at  any  point  of  contact  will  pass  through  the 
point  A  (the  pitch  pointy.      Such  curves  are  said  to  be  conjugate. 

It  is  not  necessary  that  the  describing  point  be  on  the  circumference  of  the  circle,  or  that 
the  describing  curve  be  a  circle,  in  order  to  obtain  two  curves  which,  acting  together,  shall  pro- 
duce a  constant  velocity  ratio. 


10  LAW  OF  TOOTH  CONTACT  AND  SPUR  GEARS. 

14.  Law  of  Tooth  Contact.  In  order  to  preserve  the  condition  of  constant  velocity  ratio, 
the  tooth  outlines  which  act  in  contact  must  be  such  that  the  common  normals  at  the  point  of 
contact  shall  always  cut  the  line  of  centers  in  the  same  point;  and  in  general,  the  curves  must 
be  sucli  as  may  be  simultaneously  traced  upon  the  planes  of  rotation  of  two  disks,  while  re- 
volving, by  a  marking  point  which  is  carried  by  a  describing  curve,  moving  in  rolling  contact 
with  both  disks. 

15.  Application.  Suppose  action  to  take  place  between  the  odontoids,  or  gear  teeth,  shown 
in  Fig.  2,  Plate  3.  Let  1  be  the  driver,  and  suppose  motion  to  begin  from  the  position  shown 
in  the  figure,  the  contact  being  at  A,  As  the  motion  takes  i)lace,  points  A',  A",  A'",  will  succes- 
sively come  into  contact,  their  common  normals  passing  through  the  PITCH  point.  A,  at  the 
time  of  their  contact,  thus  producing  a  constant  velocity  ratio,  and  the  periphery,  or  pitch 
LINE,  of  1,  will  have  the  same  velocity  as  the  j)eriphery,  or  pitch  line,  of  2.  But  this  uniform 
motion  must  cease  when  points  A'"  A'"  come  into  contact,  and  the  velocity  ratio  will  remain  con- 
stant no  longer,  unless  a  second  pair  of  curves  begin  contact  at  this  moment. 

Plate  4  illustrates  a  pair  of  disks  provided  with  a  series  of  these  curves  arranged  so  as  to 
continue  the  motion  indefinitely  in  either  direction. 

16.  Spur  Gears.  Plate  4.  F  is  the  center  of  a  pinion  having  twelve  teeth,  and  G  the 
center  of  a  gear  of  eighteen  teeth,  only  a  segment  of  the  latter  being  shown.  A  C  K  is  the 
describing  circle,  carrying  the  marking  point  C,  wliich  described  the  epicycloid  C  D,  and 
the  hypocycloid  C  E.  The  depth  of  the  pinion  tooth  must  be  made  sufficient  to  admit  the 
addendum  of  the  gear  tooth,  but  only  that  portion  of  the  curve  between  C  and  E  will  engage 


CIRCULAR    AND    DIAMETER    PITCH.  11 

C  D.  The  reiiiiiiiider  of  the  pinion  flank  may  be  a  continuation  of  the  Iiypocych>id,  or  any 
otlier  curve  which  may  not  interfere  with  the  action  of  the  gear  tooth.  The  opposite  sides  of 
the  teeth  are  unulc  alike  in  order  that  motion  may  take  pkice  in  cither  direction.  If  tlie  direc- 
tion l)e  that  indicated  hy  the  arrows,  tiie  pinion  hcing  the  driver,  the  .shaded  side  of  the  teeth 
would  have  contact ;    and  if  the  direction  he  reversed  the  opposite  faces  would  engage. 

In  order  to  accurately  reproduce  the  dedenda  of  the  pinion,  a  scroll  may  be  used  in  the  fol- 
lowing manner : — 

Having  selected  one  to  match  the  tooth  curve,  C  E,  continue  the  curve  of  the  scroll  l>y  the 
center  F,  from  which  a  circle  should  be  drawn  tangent  to  the  line  of  the  scroll.  ]Mark  that 
point  of  the  scroll  in  contact  with  the  pitch  circle.  Having  laid  off  the  pitch,  and  thickness  of 
the  teeth,  place  the  marked  point  of  the  scroll  to  coincide  with  these  points,  and  at  the  same 
time  tangenl,  to  the  circle  already  drawn.  Draw  such  part  of  the  curve  as  lies  between  the 
addendum  and  dedendum  circles.  Reverse  the  scroll  for  drawing  the  opposite  side  of  the 
teeth. 

17.  Circular  Pitch.  The  distance  A  D,  or  A  E,  measured  on  the  i)itch  line  between  cor- 
responding  jjoints   of    consecutive   teeth,    is    called   the    circular    pitch,    and    is    equal   to   the 

circumference  of  pitch  circle 
number  of  teeth 

Let  P'  denote  the  circular  pitch,  D'  the  diameter  of  the  pitch  circle,  and  N  the  number  of 
teeth,  then  will  P'  =  '^  (1),  and,  ~,  =  ^,  (2). 

18.  Diameter  Pitch.  Tn  order  to  express  in  a  more  direct  and  simple  manner  the  ratio 
between  the  diameter  of  the  pitch  circle  and  the  number  of  teeth,  and  to  easily  determine  the 


12  TOOTH    PARTS. 

proportions  of  the  tee.th,  it  has  been  found  expedient  to  apply  the  term  pitch,  or  more  properly, 
diameter  pitch,  to  designate  the  ratio  between  the  number  of  teeth  and  tlie  diameter  of  pitch 
circle.  This  is  7iot  an  ahsoJufe  measure,  but  a  ratio  ;  and  since  it  may  usually  be  expressed  by  a 
whole  number,  the  proportions  of  the  parts  of  a  tooth,  which  are  commonly  dependent  on 
the  pitch,  may  be  more  readily  determined,  and  all  the  figuring  of  the  gear  simplified. 

Designating  the  diameter  pitch  by  P,  P  =  q"/  (3). 

To  obtain  the  relation  between  the  diameter  pitch  and  the  circular  pitch,  compare  formulas 
2  and  3.  -,  =  —,,  ^,  =  P  I  hence  5;  =  P  or  P  P'  =  tt  (4).  This  last  equation  expresses  the  relation 
between  the  two  pitches  in  a  simple  form  which  may  be  easily  remembered. 

Illustration.  —  The  pinion  represented  in  Plate  4  has  12  teeth,  and  is  3  inches  in 
diameter.  5;  =  P,  ^  ^ '^'  ^^^  pitch,  therefore,  is  4.  The  circular  pitch,  P'  =  ^  =  — ^  =  -7854. 
Having  given  any  two  of  the  terms,  N ,  D',  P,  P',  the  other  terms  may  be  determined. 

19.  Face,  or  Addendum.  That  portion  of  the  tooth  curve  lying  outside  of  the  pitch  circle 
is  called  the  face  or  addendum,  as  C  D,  Plate  4. 

20.  Flank,  or  Dedendum.  That  portion  of  the  tooth  curve  lying  inside  of  the  pitch  circle 
is  called  the  flank  or  dedendum,  as  E  H ,  Plate  4. 

21.  Path  of  Contact.  In  Fig.  1,  Plate  3,  it  will  be  observed  that  the  contact  between 
the  two  curves  takes  place  in  the  arc  A  A'  A"  A'".  This  is  called  the  path  of  contact,  or  line 
of  action,  and  in  the  c^xloidal  system  this  line  is  an  arc  of  the  describing  circle. 


ARCS    OF    CONTACT.  lo 

22.  Arc  of  Contact.  The  arc  described  by  a  point  on  the  pitch  line  during  the  time  of  con- 
tact of  two  odontoids  is  called  the  arc  of  contact.  It  must  not  be  less  than  the  pitch.  In  this 
case  the  arc  of  contact  would  Ije  measured  by  the  arcs  A  D  or  A  E ,  and  these  arcs  being  equal  to 
the  pitch,  the  case  is  called  a  limiting  one.  In  practice  it  should  be  greater,  which  would  be 
accomplished  by  lengthening  the  addendum.     Plate  4. 

23.  Arcs  of  Approach  and  Recess.  There  are  four  cases  of  contact  that  may  take  place 
between  the  gear  and  pinion  of  Plate  4. 

1.  Gear  as  driver.     Direction  opposed  to  the  arrows.     Contact  begins  at  A  and  ends  at  C. 

2.  Pinion  as  driver.     Direction  same  as  arrows.     Contact  begins  at  C  and  ends  at  A. 

In  each  of  these  cases  the  action  will  take  place  between  the  shaded  portions  of  the  teeth. 

3.  Gear  as  driver.     Direction  same  as  arrows.     Contact  begins  at  A  and  ends  at  L. 

4.  Pinion  as  driver.     Direction  opposed  to  the  arrows.     Contact  begins  at  L  and  ends  at  A. 
In  the  last  two  cases  there  will  be  no  contact  between  the  shaded  portions  of  the  teeth. 

In  the  first  and  third  cases  the  contact  takes  place  from  the  pitch  point,  and  the  arc 
described  by  a  point  on  the  pitch  line  during  tliis  action  is  called  the  arc  of  recess. 

In  the  second  and  fourth  cases  the  contact  takes  place  toward  the  pitch  point,  ending  at  A, 
and  the  arc  described  by  a  point  on  the  pitch  line  during  this  action  is  called  the  arc  of  approach. 

It  should  also  be  observed  that  in  the  case  illustrated  the  arc  of  contact  must  be  either  one 
of  approach  or  of  recess;  but  had  the  teeth  of  each  gear  been  provided  with  curves  on  both 
sides  of  the  pitch-line,  as  in  Plate  5,  the  arc  of  contact  would  have  consisted  of  an  arc  of 
approach  and  of  recess.  (See  Art.  30,  page  10,  for  a  further  discussion  of  the  relation  between 
these  arcs.) 


14  ANGLE  OF  PRESSURE. RACK. 

24.  Angle  of  Obliquity,  or  Pressure.  The  angle  which  the  common  normal  to  a  pair  of 
conjugate  teeth  makes  with  the  tangent  at  the  pitch  point,  is  called  the  angle  of  obliquity,  or 
angle  of  pressure.  The  angle  CAP,  Plate  4,  is  the  angle  of  greatest  obliquity.  The  greater 
this  angle,  the  greater  the  tendency  to  thrust  the  gears  apart ;  the  friction  will  be  increased 
and  the  component  of  force  tending  to  produce  rotation  will  be  decreased. 

25.  Rack.  If  the  diameter  of  the  gear  be  indefinitely  increased,  the  pitch  circle  will  finally 
become  a  right  line,  and  the  gear  will  then  be  known  as  a  rack. 

The  rack  shown  in  PLx^te  4  has  teeth  only  on  one  side  of  the  pitch  line,  like  the  pinion  and 
gear,  and  the  conditions  of  action  are  simikir.  The  tooth-curve  will  be  a  cycloid,  and  the 
rolling  circle,  M  N  0,  must  be  the  same  as  that  used  for  the  engaging  pinion,  in  order  to  fulfil 
the  general  law  for  maintaining  a  constant  velocity  ratio  (Art.  14,  page  10). 

26.  Spur  Gears  having  action  on  both  sides  of  the  Pitch  Point,  Plate  5.  If  we  assume  the 
diameters  of  pitch  and  rolling  circles  to  be  the  same  as  before,  and  the  arc  of  action,  C  A,  un- 
changed, the  addendum  of  gear  and  dedendum  of  pinion  will  be  the  same  as  those  of  Plate  4. 
This  case,  liowever,  differs  from  the  preceding  in  tliat  the  number  of  teeth  is  but  lialf  as  great, 
and  therefore  the  pitch  will  be  doubled.  This  will  require  tlie  arc  of  action  to  be  doubled,  in 
order  that  it  shall  equal  the  pitch  (Art.  22,  page  13).  Such  increase  in  the  arc  of  action  may 
be  made  b}'  continuing  the  path  of  contact  to  the  other  side  of  the  pitch  point,  following  the 
circumference  of  a  rolling  circle  which  may  or  may  not  be  equal  to  the  other  rolling  circle. 
Having  laid  off  the  arc  A  H  equal  to  one-half  the  circuhir  pitch,  describe  the  curves  H  K  and 
H  L ,  with  H  as  the  generating  point  of  the  new  rolling  circle.     The  former  of  these  curves  will 


CURVE  OF  LEAST  CLEARANCE.  15 

b«!coiiic  the  addiMidum  of  tin;  piuiuii,  and  tliu  latter  the  dedendum  of  tlie  gear  tooth.  The  en- 
gaging gears  Mill  then  lune  l)oth  faces  and  flanks,  the  action  will  begin  at  C  and  end  at  H,  the 
path  of  contact  will  be  C  A  H,  the  arc  C  A  being  tlie  path  of  approach,  and  A  H  the  path  of 
recess,  their  snm  being  ecpial  to  the  circular  pitch. 

In  a  similar  manner  the  dedendum  of  the  rack  tooth  may  be  described  to  engage  the  adden- 
dum of  the  pinion  tooth,  and  the  contact  begun  at  N  will  end  at  0,  N  M  being  the  path  of 
approach,  and  M  0  the  path  of  recess.  That  portion  of  the  dedendum  of  rack  tooth  which 
engages  the  addendum  of  the  pinion  is  indicated  by  sectioning,  but  it  is  necessary  to  continue 
the  dedendum  to  a  depth  sufficient  to  allow  the  addendum  of  the  engaging  tooth  to  enter. 

27.  Clearance.  The  space  between  the  addendum  circle  of  one  gear  and  the  dedendum 
circle  of  an  engaging  gear  is  called  clearance.      Fig.  9,  page  17. 

28.  Curve  of  Least  Clearance.  If  the  pitch  circle  of  tlie  gear  be  rolled  on  that  of  the 
pinion,  and  the  epitrochoid  of  the  liighest  point,  C,  of  the  gear  tooth  l)e  determined,  it  will  l>e 
the  curve  of  least  clearance. 

The  successive  positions  of  the  tooth,  \\  hen  so  revoh cd,  ai-e  shown  by  the  dotted  line  in 
Plate  5,  and  tlie  line  connecting  these  points  would  be  the  desired  curve.  This  may  be 
obtained  as  follows:  .Vssume  any  point,  R  on  the  pitch  circle  of  pinion,  and  lav  off  arc  A  R'  on 
the  pitch  circle  of  gear,  equal  to  arc  A  R.  From  R ,  with  radius  R  C,  equal  to  R'  C ,  describe  an 
arc.  Similarly  deseri])e  other  ares,  and  draw  a  curve  toui'hing  these  arcs  on  the  inside.  This 
curve  will  be  the  curve  of  least  clearance.* 

*  See  also  the  method  of  Art.  71,  page  03. 


16  CONDITIONS    GOVERNING    THE    PRACTICAL    CASE. 

29.  Backlash.  In  order  to  allow  for  unavoidable  inaccuracies  of  workmanship  and  operat- 
ing, it  is  customary  to  make  the  sum  of  the  thickness  of  two  conjugate  teeth  something  less 
than  the  circular  pitch.     This  insures  contact  between  the  engaging  faces  only. 

30.  Conditions  governing  the  practical  case.  From  a  consideration  of  the  foregoing  limiting 
cases,  the  following  principles  are  deduced,  to  which  are  also  added  the  limitations  and  modifi- 
cations established  by  practice. 

1.  The  curves  of  gear  teeth,  which  act  to  produce  a  constant  velocity  ratio,  must  be 
described  by  the  same  curve  rolling  in  contact  with  their  respective  pitch  circles.  (Art.  14, 
page  10.)  Practical  considerations  limit  the  diameter  of  the  describing  circle  to  a  maximum 
of  about  -z-5  or  equal  to  the  radius  of  the  pitch  circle,  and  a  minimum  of  about  1|  P',  or  -^. 
See  also  Art.  32,  page  19. 

2.  The  arc  of  contact  must  equal  the  circular  pitch,  and  in  practice  exceed  it  as  much  as 
possible. 

3.  The  addendum  of  a  gear  tooth  engages  the  dedendum  of  the  pinion,  and  the  action 
between  them  either  l)egins  or  ceases  at  the  pitch  point. 

Since  the  addendum  and  dedendum  of  any  tooth  are  independent  curves,  they  may  be 
described  by  rolling  circles  differing  in  diameter. 

4.  In  the  limiting  cases  considered,  the  height  of  the  tooth  is  dependent  on  the  arc  of  con- 
tact, but  in  practice,  the  arc  of  contact  is  made  dependent  on  the  height  of  the  tooth. 

While  it  is  an  almost  universal  custom  to  make  the  addenda  of  engaging  teeth  equal,  there 
are  special  cases,  in  which  very  smooth-running  gears  are  required,  where  it  would  be  advan- 
tageous to  make  the  addenda  of  the  driver  less  than  those  of  the  driven  gear,  thus  increasing 
the  arc  of  recess,  or  decreasing  the  arc  of  approach. 


PROPORTIONS    OF    STANDARD    TOOTH. 


17 


The  approaching"  action  being  more  detrimental, 
by  reason  of  the  friction  induced,  it  is  common  to  de- 
sign clock  gears  so  as  to  eliminate  this  by  providing 
the  driver  with  faces  only,  and  the  driven  with  flanks 
only.  Or,  if  the  gears  are  made  with  both  faces  and 
flanks,  to  so  round  the  faces  of  the  driven  gear  that  no 
action  may  take  place. 

31.  Proportions  of  Standard  Tooth.  The  propor- 
tions most  commonly  accepted  for  cut  gears  are  those 
illustrated  in  Fig.  9.  The  dimensions  are  made  depen- 
dent on  the  pitch,  as  follows :  — 


Addendum,  (S) 
Dedendum,    (S  +  f)  = 


diam. pitch 
I 


diam  pitch 


-f-  clearance 


+  f. 


Thickness,   (t)  =  ^  circular  pitch  =  y  =  ^ 


Clearance,  (f)  =  -addendum  =  |^ 


-^  or,  f  =  -  thick- 


t        P' 
ness  of  tooth  =  ^  =  ^ 


20   P 


In    assuming   this   value   for  the  thickness   of  the 
tooth  the  backlash  is  taken  as  zero,  but  of  course  the 


Fig.  9. 


18  INFLUENCE    OF    THE    ROLLING    CIPvCLE. 

tooth  must  be  slightly  smaller  than  the  space  to  permit  of  freedom  hi  action.  If  there  be 
any  backlash  the  value  of  t  will  be  '"'"'"^"  '"^"^ ~ '^""'^'''^^ .  In  rough  cast  gears  the  backlash  may 
be  as  great  as  j^^jth  the  circular  pitch,  but  this  amount  is  very  evcessive.  It  is,  however,  in- 
consistent to  base  the  values  for  backlash,  or  clearance,  on  the  i)itch,  since  an  increase  in  tlie 
size  of  the  tooth,  or  pitch,  does  not  necessarily  mean  a  proportional  increase  in  the  allowance 
to  be  made  for  the  inaccuracies  of  workmanship.  Indeed,  both  these  clearances  must  be  left  to 
the  judgment  of  the  designer. 

Fillets.  The  circular  arc  tangent  to  the  flank  and  dedendum  circle  is  called  the  fillet.  It 
is  designed  to  strengthen  the  tooth  by  avoiding  the  sharp  corner  at  the  root  of  the  tooth.  A 
good  rule  is  that  of  making  the  radius  of  fillet  equal  to  one-seventh  of  the  space  between  the 
teeth,  measured  on  the  addendum  circle,  as  in  Fig.  9.  The  limit  of  size  may  be  determined  by 
obtaining  the  curve  of  last  clearance.     Art.  28,  page  15. 

32.  Influence  of  the  Diameter  of  the  Rolling  Circle  on  the  Shape  and  Efficiency  of  Gear 
Teeth.  If  the  height  of  the  teeth  be  previously  determined,  any  increase  in  the  diameter  of  the 
describing  circle  will  increase  the  path  of  contact  and  decrease  the  angle  of  pressure.  But 
since  an  increase  in  the  diameter  of  the  describing  circle  produces  a  weaker  tooth,  by  reason  of 
the  undercutting  of  the  flank,  as  shown  in  Fig.  12,  page  21,  the  maximum  limit  of  the  diameter 
is  commonly  made  equal  to  the  radius  of  the  pitch  circle  within  which  it  rolls.  As  was  shown 
in  Art.  9,  page  5,  this  will  generate  a  radial  flank.  In  the  case  of  geai"S  designed  to  trans- 
mit a  uniform  force,  and  not  subjected  to  sudden  shocks,  it  is  desirable  that  the  teeth  have 
radial  flanks,  and  consequently  the  diameters  of  the  rolling  circles  will  be  equal  to  the  radii 
of  the  pitch  circles  within  which  the}-  roll.      If  the  force  to  be  transmitted  be  irregular,  and  the 


INFLUENCE    OF    THE    ROLLING    CIRCLE. 

/ 

teeth  required  to  sustain  suddeu  strains,  it  is  better  that 

the  flank  be  made  wider  at  the  dedenduni  circle,  and  a 
describiug  circle  chosen  of  a  diameter  sufficiently  small 
to  produce  the  desired  result. 

In  general,  the  diameters  of  the  describing  circles 

D'  5 

will  lie  between  the  values  of  -7^  and  - .      The  second 

value  was  used  for  the  describing  circle  of  the  efears  in 
Plate  5,  and  would  describe  radial  flanks  for  a  gear 
having  ten  teeth. 

Fig.  10  illustrates  the  effect  of  a  change  in  the  di- 
ameter of  the  rolling  circle  on  the  path  of  contact  and 
angle  of  pressure.  Two  gears  of  equal  diameter  are 
su[)posed  to  engage,  and  the  teeth  are  described  by  roll- 
ing circles  of  equal  diameter. 

K  P  is  the  addendum,  and  P  L  the  dedendum  of  the 
tooth  described  by  the  rolling  circles  C  P,  and  P  D, 
which  are  of  the  same  diameter,  and  equal  to  one- 
quarter  of  the  pitch  diameter.  A  C  being  the  ad- 
dendum line  of  the  engaging  gear,  C  may  be  considered 
as  the  first,  and  D  as  the  last,  point  of  contact.  Tlie 
arcs  C  P  and  P  D  constitute  tlie  path  of  contact,  and  the 
angle  C  P  H  is  the  angle  of  pi'cssure. 

Next  consider  the  describing  circle  as  increased,  its 


Fig.  10. 


20 


INTERCHAXGEABLE    GEARS. 


91 

4 

D' 
2 

38   P' 

1.35   P'. 

32.3° 

20.3°. 

Fig.  11. 


diameter  being  equal  to  one-half  of  the  diameter  of  the  pitch 
circle.  The  form  of  the  tooth  ^vill  now  be  E  P  F,  and  the  path 
of  contact  A  P  B.  In  the  latter  case  the  arc  of  contact  will 
be  greater,  the  maximum  angle  of  pressure  less,  and  the 
tooth  weaker  than  in  the  former. 

The  relation  between  the  two  cases  may  be  more  exactly 
stated  as  follows :  — 

Diameter  of  describing  curve. 

Arc  of  contact, 

Maximum  angle  of  pressure. 

Again,  the  weakness  of  the  tooth  in  the  second  case  may 
be  partially  overcome  by  reducing  the  height  of  the  tooth, 
and  in  general  this  would  be  advantageous,  the  so-called 
standard  tooth  being  too  high  for  the  best  results. 

33.  Interchangeable  Gears.  Since  the  same  diameter  of 
rolling  circle  must  be  used  for  the  addendum  of  pinion  tooth, 
and  tlie  dedendum  of  engaging  gear  tooth,  it  follows  that  for 
any  system  of  interchangeable  gears,  the  addenda  and  de- 
denda  of  all  teeth  must  be  described  by  the  same  describing 
curve.  It  is  also  necessary  that  the  pitch,  and  proportion 
of  the  teeth,  be  constant. 


PRACTICAL    CASE. 


21 


In  practice,  it  is  common  to  regard  gears  of  twelve  or  of 
fifteen  teeth  as  tlie  base  of  the  system,  and  the  diameter  of 
the  rolling  circle  is  made  equal  to  the  radius  of  the  corre- 
sponding pitch  circle,  thus  describing  teeth  w  ith  radial  flanks 
for  the  smallest  gear  of  the  set.  If  twelve  be  adopted  as  the 
smallest  number  of  teeth  in  the  system,  the  diameter  of  the 


N 


12 


pitch  circle  will  be    D'  =  -  =  -— ,  and  the  diameter  of  the  de- 
scribing circle  will  be  ^  =  □• 

Again,  if  a  fifteen-toothed  gear  be  used  as  the  base  of 

7  5 

the  system,  the  diameter  of  the  describing  circle  will  be  -^. 

Figs.  11  and  12  illustrate  a  fifteen  and  a  nine-toothed 
gear  engaging  a  rack.  The  diameter  of  the  rolling  circle  by 
which  the  teeth  were  described  is  -^,  which  will  equal  3.75 
inches  for  a  2  pitch  gear. 

The  fifteen-toothed  gear  will  have  radial  flanks,  but  the 
nine-toothed  gear  will  have  the  flanks  much  undercut  by 
reason  of  the  diameter  of  the  rolling  circle  exceeding  the  ra- 
dius of  the  pitch  circle. 

34.  Practical  Case  of  Cycloidal  Gearing.  Plate  6.  Let 
F  and  G  be  the  centers  of  pinion  and  gear  having  twelve  and 
eighteen  teeth  respectively,  and  a  diameter  pitch  of  4.     The 


Fig.  12. 


22  SPUR    GEARS. 

pitcli  diameters  will  equal  p  =  t  =  ^  inches,  and  □  =-^  =4i^  inches  (Art.  18,  page  12).     If  the 

tooth  he  of  standard  dimensions,  the  addendum  and  dedendnm  lines  may  he  determined  and 
drawn  hj  Art.  31,  page  17.  The  diameter  of  the  rolling  circle  is  assumed  to  he  1^-  inches  for 
the  addendum  and  dedendum  of  hoth  gears.  Since  the  teeth  should  usually  he  shown  in  con- 
tact at  the  pitch  point,  suppose  the  generating  point  of  the  describing  curve  to  he  at  this  point, 
and  describe  the  curves  by  rolling  the  circles  from  this  position,  first  on  the  inside  of  one  pitch 
circle,  and  then  on  the  outside  of  the  other  pitch  circle,  thus  obtaining  the  flank  of  one  tooth, 
and  the  engaging  face  of  a  tooth  of  the  other  gear. 

An  enlarged  representation  of  these  curves  is  shown  in  Plate  6.  They  may  be  drawn  by 
the  methods  of  Arts.  6  and  7,  page  5,  or  by  Art.  9,  page  5,  but  care  should  be  used  to  draw 
them  in  their  proper  relation  to  each  other,  as  shown  in  the  figure,  so  that  it  may  not  be  neces- 
sary to  reverse  the  curves  in  order  to  incorporate  them  into  tooth  forms.  The  order  for  the 
drawing  of  the  curves  may  be  A  B ,  A  T,  A  D ,  A  S. 

Instead  of  reproducing  the  tooth  curves  by  means  of  scrolls,  it  is  sufficiently  accurate,  and 
much  more  rapid,  to  approximate  them  by  circular  arcs.  Plate  2,  Fig.  3,  illustrates  a  simjjle 
method  which  closely  approximates  the  curves  of  this  system,  and  suffices  for  the  ordinary 
drawing  of  a  gear,  but  in  no  case  should  be  used  for  describing  the  curves  for  a  templet.  This 
method  consists,  frst,  in  the  construction  of  a  normal  for  a  point  of  the  curve  at  a  radial 
distance  from  the  pitch  line  equal  to  two-thirds  of  the  addendum  or  dedendum  of  the  tooth; 
second,  in  the  finding  of  a  center  on  this  normal,  such  that  an  arc  ma}"  be  described  through 
the  pitch  point,  and  the  point  of  the  tooth  already  determined.  A  P  is  the  height  of  the  adden- 
dum,  and  B  a  point  radially  distant  from  the  pitch  line,  equal  to  -  A  P,  through  which  the  arc 


PRACTICAL    CASE.  23 

B  E  is  drawn.  When  the  point  E  of  the  descn])ing  curve  sliall  have  become  a  point  of  contact, 
as  at  E',  the  arc  E'  P  being  erjual  to  E  P,  the  point  P  Avill  have  moved  to  T,  the  chord  T  E'  being 
equal  to  the  chord  E  P.  T  will  })e  a  point  in  the  addendum,  and  T  E'  the  normal  for  this  point. 
From  a  point,  M,  on  this  normal,  and  found  by  trial,  describe  the  arc  P  T,  limited  by  the  ad- 
dendum line.     Similarly  the  curve  of  the  dedendum  may  be  determined. 

Having  determined  such  centers  as  may  be  required  for  describing  the  tooth  curves,  di-aw 
circles  through  these  centers,  as  indicated  in  Plate  6,  to  facilitate  the  drawing  of  other  teeth. 
The  radius  for  the  dedendum  is  often  inconveniently  great,  and  in  such  cases  it  is  desirable  to 
use  scrolls,  employing  the  method  of  Art.  1(],  page  11.  Next  divide  the  pitch  circle  into  as 
many  equal  parts  as  there  are  teeth,  beginning  at  the  pitch  point.  From  each  of  these  divisions 
lay  off  the  thickness  of  the  teeth.  If  there  be  no  backlash,  this  thickness  will  equal  one-half 
the  circular  pitch  ;   Ijut  if   an  amount  be  determined  for  backlash,  the  thickness  will  equal 

P^—  backlash 
2 

The  circle  of  centers  liaving  been  drawn,  the  tootli  curves  should  be  described.  These  will 
be  limited  l)y  tlie  addendum  and  dedendum  circles  already  drawn.      Finally  draw  the  fillets. 

The  maxinuim  angle  of  pressure  between  the  pinion  and  gear  will  be  24°,  the  arc  of 
approach  .52,  the  arc  of  recess  .48,  and  their  sum,  Avliich  is  the  arc  of  contact,  1  inch,  or  1.27 
times  the  circular  ])itch. 

The  rack  teeth  would  be  similarly  described.  The  })itch  line  l)einga  right  line,  the  circular 
pitch  may  be  laid  off  directly  by  scale,  or  spaced  from  the  pinion.  The  approximate  method 
may  be  used  for  the  tooth  curves,  and  lines  drawn  parallel  to  the  pitch  line,  for  the  centers  of 
the  arcs  which  approximate  the  addenda  and  dedenda  of  the  teeth. 


24 


GEAR    FACE. 


' 

y/w, 

P 

m 

m 

m 

^////., 

'mmm 

^ 

^ 

,_>    I, 



Fig.   14. 


Fig.   15. 


35.  Face  of  Gear.  In  the  previous  consideration  of 
gear  teetli  no  attention  has  been  paid  to  the  width  of  the 
gear,  or,  as  it  is  commonly  termed,  the  face  of  the  gear. 
This  dimension  is  one  of  the  factors  to  be  considered  in 
determining  the  strength  of  the  tooth,  which  is  a  subject 
apart  from  the  kinematics  of  gearing.  It  should  be  ob- 
served, however,  that  the  tooth  having  appreciable  width, 
must  be  generated  by  an  element  of  a  rolling  cylinder  in 
place  of  the  point  of  a  rolling  circle. 

36.  Comparison  of  Gears,  illustrated  in  Plates  4,  5, 

and  6.  In  the  three  cases  previously  considered,  the  di- 
ameter of  the  pitch  circles  are  equal,  and  only  one  diam- 
eter of  rolling  circle  has  been  used. 

In  Plates  4  and  5  the  arc  of  contact  is  equal  to  the 
circular  pitch ;  but  the  pitch  of  the  latter  is  twice  as  great 
as  the  former,  hence  there  are  but  half  as  many  teeth.  In 
Plate  6  the  arc  of  contact  is  made  dependent  on  the 
height  of  the  tooth,  which  is  a  standard  so  chosen  as  to 
permit  of  an  arc  of  contact  sufficiently  long  for  a  practical 
case.  But  in  Plates  4  and  5  the  height  of  the  tooth  is 
dependent  on  the  arc  of  contact,  which  latter  is  made  the 
least  possible. 


CONVENTIONAL    REPRESENTATION    OF    SPUR   GEARS.  25 

The  number  of  teeth  in  the  pinions  of  Plates  4  and  6  is  the  same;  but  in  the  former  the 
action  is  only  on  one  side  of  the  pitch  point,  there  being  no  addenda  to  the  teeth,  hence  the 
limited  arc  of  contact. 

In  Plates  4  and  5  there  is  contact  between  only  one  pair  of  conjugate  teeth,  save  at  the 
instant  of  beginning  and  ending  contact ;  while  in  the  case  of  Plate  6,  two  pairs  of  conjugate 
teeth  may  be  in  contact  during  a  part  of  the  arc  of  contact. 

37.  Conventional  Representation  of  Spur  Gears.  In  making  drawings  of  gears,  it  is  usually 
best  to  represent  them  in  section,  as  in  Fig.  14.  This  enal)les  one  to  give  complete  informa- 
tion concerning  all  details  of  the  gear,  save  the  character  of  the  teeth.  If  the  latter  be  special, 
an  accurate  drawing  of  at  least  two  teeth  and  a  space  will  be  required.  Should  it  be  neces- 
sary to  represent  the  geai-s  on  the  plane  of  their  pitch  circles,  as  in  Plate  6,  they  may  be  shown 
as  in  Fig.  13,  thus  avoiding  the  representation  of  the  teeth.  Again,  if  it  be  necessar}'  to  show 
a  full  face  view  of  the  gears,  the  method  illustrated  in  Fig.  15  may  be  employed  to  advantage. 
This  is  simply  a  S3^stem  of  shading ;  and  no  attempt  is  made  to  represent  the  proper  number  of 
teeth,  or  to  obtain  their  projection  from  another  view. 


26  INVOLUTE    SYSTEM. 

CHAPTER   IV. 

INVOLUTE     SYSTEM. 

38.  Theory  of  Involute  Action.  If  the  describing  curves  be  other  than  circles  we  shall  obtain 
odontoids  differing  in  character  from  those  already  studied ;  but  so  long  as  both  pinion  and 
gear  are  described  by  the  same  rolling  curve,  the  velocity  ratio  will  remain  constant.  The 
class  of  odontoids  ilhistrated  l)y  Plate  7,  Fig.  1,  is  known  as  the  involute,  or  single-curve 
tooth.  This  curve  cannot  be  described  by  rolling  circles,  but  may  be  generated  by  a  special 
curve  rolling  in  contact  with  both  pitch  surfaces.*  But  as  the  curve  may  be  described  by  a 
much  more  simple  process,  the  above  statement  is  of  interest  onl}-  as  showing  the  conformity 
of  the  curve  to  the  general  law.     (Art.  14,  page  10.) 

F  and  G,  Plate  7,  Fig.  2,  are  the  centers  of-  two  disks  designed  to  revolve  about  their 
respective  axes  with  a  constant  velocity  ratio,  which  is  maintained  in  the  following  manner:  — 
Suppose  the  disks  to  be  connected  by  a  perfectly  flexil)le  and  inextensible  l)and,  D  C  B  A,  wliich 
being  wound  on  the  surface  of  one,  will  be  unwound  from  the  other,  after  the  manner  of  a 
belt,  producing  an  equal  circumferential  velocity  in  the  disks.  Conceive  a  marking  point  as 
fixed  to  the  band  at  A,  so  that  during  the  motion  from  A  to  D,  curves  may  be  described  on  the 
extensions  of  disks  1  and  2,  in  a  manner  similar  to  that  described  for  the  generating  of  the 
cycloidal  curves.      When  the  point  A,  on  the  band,  shall  have  moved  to  B,  the  curve  X^  B  will 

*  For  description  of  this  metliod,  see  MacCord's  Kinematics,  page  105,  AuT.  279. 


THE    INVOLUTE    CURVE.  27 

Juive  Ik'C'H  described  on  the  extension  of  disk  2,  and  B  Aj,  on  that  of  disk  1.  Wlien  the  motion  of 
the  nuuking-  point  sliall  have  continued  to  C,  X2  Yj  C  will  have  been  described  on  the  extension 
of  disk  2,  and  A.^  B^  C,  on  that  of  disk  1.  Finally,  when  the  maiking  point  shall  have  reached 
D,  the  curve  X3  ¥3  Z^  D  will  ha\e  lieen  described  on  the  extension  of  disk  2,  and  A3  B.,  C^  D  on  the 
extension  of  d'sk  1. 

If  these  curves  be  made  the  outlines  of  gear  teeth,  and  the  former  act  against  the  latter 
so  as  to  produce  motion  opposed  to  that  indicated  by  the  arrows,  a  uniform  velocity  ratio  will 
be  maintained  between  the  disks.  On  investigation,  these  curves  will  be  found  to  be  involutes, 
A3  D  being  an  involute  of  the  periphery  of  disk  1,  and  Xg  D,  an  involute  of  disk  2.  The  curves 
may,  therefore,  be  described  by  the  method  for  drawing  an  involute  (Airr.  12,  page  7),  the 
path  of  contact,  A  D,  being  spaced  off  on  the  base  circle  from  A  to  Ag.  and  the  involute  drawn 
from  A3;  or  the  line  A  D  may  be  conceived  as  wrapped  about  the  base  circle  beginning  the  curve 
at  D. 

39.  Character  of  the  Curve.  Plate  7,  Fig.  1,  rejiresents  the  involute  curve  of  Fig.  2 
incorporated  into  gear  teeth.  It  becomes  necessary  to  continue  the  line  of  the  tooth  within  the 
periphery  of  the  disk,  which  wmII  now  be  designated  as  the  base  circle,  so  as  to  admit  the 
addenda  of  engaging  teeth.     This  portion  of  the  tooth  is  made  a  radial  line. 

The  pitch  point  being  at  B,  (the  intersection  of  the  line  of  centere  and  the  line  of  action), 
the  pitch  circles  will  be  drawn  through  this  point. 

The  circles  from  which  the  invf>lute  curves  are  described,  are  called  base 

Base  Circle  Defined.  ,  ,    .      .•,.  ,  ,  .  ,,  iiT 

nwr/es.      1  heir  tliametei"s  bear  the  same  ratio  to  each  other  as  do  the  diam- 
eters of  the  pitch  circles. 


28  CHARACTER  OF  THE  INVOLUTE. 

The  Path  ot  Contact  '^^^'^  ^"^®  ^^  actioii,  01'  ptitli  of  coiitact,  is  a  right  line  tangent  to  the  base  cir- 

a  Right  Line.         qIq^^   j^  i,s  the  line  followed  by  the  marking  point  of  the  model.   Plate  7,  Fig.  2. 

Since  the  path  of  contact  is  a  right  line,  and  as  the  common  normals  at  the  point  of  con- 

constant  Angle  of      ^'^^^  miist  alwajs  pass  through  the  pitch  point  (Art.  14,  page  10),  it  fol- 

pressure.  j^^yg  that  the  line  of  pressure,  or  angle  of  the  normals,  is  constant. 

The  action  between  the  teeth  of  the  gears  in  Fig.  1,  begins  at  D,  and  ends  at  A,  taking 
Limit  of  Action        placc  oulj  betwecu  the  points  of  tangency  of  the  line  of  action  and  base 
circle.     No  involute  action  can  take  place  2vithin  the  base  circles. 

If  the  distance  between  the  centers  of  the  gear  be  increased  or  decreased,  the  angle  of  pres- 
sure, and  length  of  the  path  of  contact  will  be  increased  or  decreased,  but  the  involute  curve, 
which  is  dependent  on  the  diameter  of  the  base  circle  only,  will  remain  unchanged.     Hence,  any 
An  Increase  in  the  cen-  chaugc  iu  thc  distaucc  bctwccn  the  centers  of  two  involute  gears  will  not 
ter  Distance  does  not   dianoe  the  velocitv  ratio,  if  the  arc  of  action  is  not  less  than  the  circular 

Affect  the  Velocity  &  ^  ^ 

Ratio.  pitch.     The  case  illustrated  by  Fig.  1  is  a  limiting  one  ;  and  therefore  an  in- 

crease in  the  center  distance  would  mean  an  increase  in  the  height  of  the  tootli,  in  order  that  the 
arc  of  action  should  equal  the  increased  pitch,  an  increase  in  the  center  distance  necessitating 
an  increase  in  the  diameters  of  the  pitch  circles,   and  therefore  in  the  circular  pitch.      But 
while  the  action  between  the  teeth  continued,  the  velocity  ratio  would  remain  constant. 
Since  the  angle  of  pressure  is  constant,  and  the  paths  of  the  elements  of  a  rack  tooth  are  right 
The  Involute  Rack      li^GS,  it  follows  that  the  tooth  outline  of  an  involute  rack  must  be  a  right 
tooth,  a  Right  Line.     |jj-jg^  perpcndicular  to  the  angle  of  pressure.     Plate  8  illustrates  a  rack  for 
an  involute  gear,  having  an  angle  of  pressure  of  about  30°.     (The  section  lined  portions  are 
not  involute.) 


INVOLUTE    LIMITING    CASE.  29 

40.  Involute  Limiting  Case.  Plate  8.  Let  the  cliametei's  of  the  pitch  circles,  the  angle 
of  pressure,  and  the  niinihtT  of  teeth,  be  given.  Having  drawn  the  pitch  circles  about  their 
respective  centers,  F  and  G ,  obtain  the  base  circles  as  follows  :  — 

Through  the  pitch  point,  B,  draw  A.D,  making  an  angle  with  the  tangent  at  the  pitch  point 
equal  to  the  angle  of  pressure.  This  will  be  the  line  of  action  ;  and  perpendiculars,  F  A  and 
D  G,  drawn  to  it  from  centers  F  and  G,  will  determine  the  radii  of  the  base  circles,  and  the 
limit  of  the  action,  or  path  of  contact,  at  A  and  D.  This  is  a  limiting  case,  in  that  the  path  of 
contact  is  a  maximum,  and  the  arc  of  contact  equal  to  the  circular  pitch.  Next  determine  the 
point,  C ,  by  spacing  the  arc,  D  K  C ,  equal  to  D  A ;  A  and  C  will  be  two  points  in  the  involute 
curve  of  the  base  circle,  D  K  C,  from  which  other  points  may  be  obtained.  Similarly  describe 
D  P,  the  involute  of  the  other  base  circle,  just  beginning  contact  at  D.  The  height  of  the  teeth 
will  be  limited  by  the  addendum  circles  drawn  through  D  and  A,  from  centers,  F  and  G.  The 
dedendum  circles  are  made  to  admit  the  teeth  without  clearance.  The  pinion  teeth  are  pointed, 
and  the  gear  teeth  fill  the  space,  having  no  backlash.  The  circular  pitch  may  be  found  by  divid- 
ing the  circumference  of  the  pitch  circle  into  as  many  parts  as  there  are  teeth,  or  the  teeth  may 
be  spaced  on  tlie  base  circle.* 

The  rack  is  made  to  engage  tlie  pinion  in  tlie  following  manner  :  — 

0  ])eing  the  pitch  point  of  rack  and  pinion,  tlie  riglit  line,  0  R.  drawn  through  this  point, 
and  tangent  to  the  base  circle,  will  be  the  path  of  contact  for  motion  in  the  direction  indicated 
by  the  arrow.  The  contact  will  begin  at  R  and  end  at  S,  the  latter  point  l)eing  that  of  the 
intersection  of  the  path  of  contact  and  addendum  circle.  The  rack  tooth  will  be  perpendicular 
to  the  line  of  action,  R  S ;  and  the  thickness  of  tooth  will  equal  that  of  the  gear  tooth,  there 
*  For  further  details  concerning  the  construction  of  this  pinion  and  gear,  see  Problem  4,  Page  76. 


30  EPICYCLOIDAL    EXTENSION    OF    INVOLUTE    TEETH. 

beino-  no  backlash  in  eitber  case.  The  addendum  of  the  rack  tooth  will  be  limited  by  the 
parallel  to  the  pitch  line  draAvu  through  the  first  point  of  contact,  R ;  and  the  dedendum  made 
sufficiently  great  to  admit  the  pinion  tooth  without  clearance. 

41.  Epicycloidal  Extension  of  Involute  Teeth.  Tlie  extent  of  the  involute  action  between 
the  gear  and  the  pinion  of  Plate  8  is  limited  to  the  path  D  A  ;  for  while  an  increase  in  the 
height  of  the  gear  tooth  is  possible,  the  limit  of  the  engaging  involute  tooth  is  at  A,  since  no 
part  of  an  involute  curve  can  lie  within  its  own  base  circle.  It  is,  however,  entirely  feasible 
to  continue  the  contact  by  a  cycloidal  action,  in  the  following  manner :  — 

The  angle  FAB  being  a  right  angle,  the  circle  described  on  F  B  as  a  diameter  must  pass 
through  A.  This  point  may  therefore  be  considered  as  a  point  m  an  epicycloid,  described  by 
the  rolling  circle  FAB,  and  having  A  B  for  its  normal,  which  is  also  the  normal  for  the  involute. 
But  this  diameter  of  rolling  circle  being  one-half  the  pitch  circle  within  Avhicli  it  rolls,  the 
h}-pocycloid  Avill  be  a  radial  line,  and  the  dedenda  of  the  teeth  aaIII  be  radial  within  the  base 
circle.  By  rolling  the  same  circle  on  the  outside  of  the  gear  pitch  circle,  the  addenda  of  the 
gear  teeth  may  be  extended,  and  the  path  of  contact  continued  to  N,  Avliich  is  a  limit  in  this 
case,  by  reason  of  the  gear  tooth  having  become  pointed. 

Similarly,  the  addendum  of  the  rack  tooth  ma}-  be  extended  by  the  same  describing  circle. 
In  the  figure  it  is  made  sufficiently  long  to  just  clear  the  dedendum  circle  required  for  the 
pointed  gear  tooth.  The  action  will  now  begin  at  Q,  follow  the  rolling  circle  to  R,  and  then, 
becoming  involute,  contintie  to  S. 

42.  Involute  Practical  Case,  Plates  9  and  10.  Having  given  the  number  of  teeth  of 
engaging  geai-s,  and  the  diametei-s  of  their  pitch  circles,  it  is  required  to  determine  the  curves 
for  the  involute  teeth  of  a  pinion,  gear,  and  rack. 


INVOLUTE    PRACTICAL    CASE.  31 

The  diameters  of  the  describing  circles  would  be  of  fii-st  consideration  in  cycloidal  gearing ; 
while  in  the  involute  system,  the  angle  of  pressure  or  line  of  action  must  first  be  established ; 
and  tangent  to  this  the  base  circles  may  be  drawn.  My  reference  to  Platk  7,  Fig.  1,  it  will 
be  seen  that  with  a  constant  center  distance,  a  decrease  in  the  angle  of  pressure  will  necessi- 
tate an  increase  in  the  diameter  of  the  base  circles,  and  a  corresponding  decrease  in  the  path  of 
contact.  That  is  to  sa}',  an  increase  in  the  possible  length  of  the  path  of  contact  means  an 
increase  in  the  angle  of  pressure.  In  Plate  7,  Fig.  1,  this  angle  is  too  great  for  actual  prac- 
tice, being  about  30°;  3'et  it  cannot  be  lessened  in  this  case,  as  the  number  of  teeth  is  limited. 
Practice  has  limited  this  angle  to  14.V°  or  15°,  which  is,  unfortunately,  too  small;  but  as  one 
of  these  angles  is  generally  adopted  in  the  manufacture  of  gears,  the  latter  will  be  used  in  the 
following  problem  :  — 

A  pinion  of  12  teeth  is  recpiired  to  engage  a  gear  of  30  teeth,  and  a  rack,  the  diameter 
pitch  being  1.     The  former  is  illustrated  by  Plate  9,  and  the  latter  by  Plate  10. 

Pinion  diameter  =  d'  =  -=-=12  inches. 

Gear  diameter     ^  D'  =  ^  =  I^  =  30  inches.     (Art.  18,  page  11.) 

Since  the  teeth  are  to  be  of  standard  dimensions  (Art.  31,  page  17),  the  addenda  will  be 
1  inch,  the  dedenda  li  inches;   and  there  being  no  backlash,  the  thickness  of  the  teeth  will  be 

half  the  circular  pitch,  or  ^'.     The   circular  2>itch,   p;  =^=3.141G.     Draw  the  pitcli  cin-les. 

The  line  of  action  will  jkiss  thi'ough  tlie  pitch  point,  making  the  i-iMpiired  angle  with  the 
common  tangent  at  this  point.  Next  draw  the  base  circles  tangent  to  this  line,  and  determine 
the  points  of  tangency,  D  and  A.     Construct  the  involutes  of  these  base  circles  in  the  manner 


32 


INTERFERENCE. 


Fig.  16. 


indicated  by  Fig.  16,  and  according  to  the  method  for  describ- 
ing an  involute,  Art.  12,  page  7.  It  will  now  be  seen  that 
the  gear  tooth  will  be  limited  by  the  arc  drawn  through  D, 
the  point  of  tangency  of  base  circle  and  line  of  action.  If, 
however,  the  involute  curve  be  continued  to  the  addendum 
circle,  as  shown  by  the  dotted  line,  C  E,  it  will  interfere  with 
the  radial  portion  of  the  j^inion  flank,  which  lies  within  the 
base  circle.  The  pinion  tooth  will  have  no  such  limitation, 
since  the  addendum  circle  intersects  the  line  of  action,  D  A, 
at  L,  a  considerable  distance  from  the  limit  of  involute  action, 
at  the  point  A. 

Similarly,  the  rack  tooth  will  l^e  found  to  interfere  with 
the  pinion  flank,  if  extended  beyond  the  point  C,  which 
comes  into  contact  at  the  point  D,  the  limit  of  involute  ac- 
tion. But  the  pinion  face  may  be  extended  indefinitely,  so 
far  as  involute  action  is  concerned.  The  remedy  for  this 
interference  is  treated  of  in  the  following  article. 

43.  Interference.  Since  practical  considerations  demand 
the  maintenance  of  a  standard  proportion  of  tooth,  two 
schemes  are  adopted  for  avoiding  or  correcting  this  inter- 
ference, observed  in  Plates  9  and  10. 

The  fu-st  is  to  hollow  that  part  of  the  pinion  flank  lying 


INFLUENCE    OF    THE    ANGLE    OF    PRESSURE.  33 

within  the  l)ase  circle  so  as  to  clear  the  interfering  part  of  tlie  gear,  or  rack  tooth.  In  this 
case  there  will  be  no  action  beyond  the  point  of  tangency  D.  The  second  method  consists  in 
making  the  interfering  portion  of  the  addendum  an  epicycloid  described  by  a  circle  of  a  diam- 
eter equal  to  the  radius  of  the  pinion  pitch  circle.  Such  a  describing  circle  would  generate  a 
radial  flank  for  that  part  of  the  curve  lying  within  the  base  circle.  By  this  means,  the  action 
will  be  continued  and  the  velocity  ratio  maintained,  although  the  action  will  cease  to  be 
involute.     Art.  41,  page  30. 

44.  Influence  of  the  Angle  of  Pressure.  The  interference  may  be  entirely  olniated  by 
sufficiently  increasing  the  angle  of  pressure  ;  but  in  the  case  cited  (Plates  9  and  10)  it  would 
necessitate  an  angle  of  24,1°,  which  is  too  great  for  general  use.  Had  the  number  of  teeth  in 
the  pinion  been  greater,  the  interference  would  have  been  less,  and  with  80  teeth  in  the  pinion, 
there  would  have  been  no  interference.     See  Art.  45. 

The  angles  of  14.1°  and  15°,  commonly  adopted,  are  unfortunately  small.  There  is.  how- 
ever, a  tendency  to  increase  this  angle,  and  geai"S  for  special  machines  have  been  made  with  a 
20°  angle  of  pressure.  This  latter  angle  will  permit  gears  having  18  teeth  to  engage  without 
interference,  and  the  thrust  due  to  this  increase  in  the  angle  of  pressure  is  an  insignificant 
amount.  A  system  based  on  this  angle  of  pressure  would  unquestionably  be  an  improvement 
over  the  present  one. 

45.  Method  for  determining  the  Least  Angle  of  Pressure  for  a  Given  Number  of  Teeth  having 
no  Interference  when  engaging  a  Rack.     Pig.  17. 

Let  A  be  the  center  of  a  gear  having  A  B  =  R  for  the  radius  of  pitch  circle,  and  D  B  T  the 


34 


LEAST    ANGLE    OF    PRESSURE    WITHOUT    INTERFERENCE. 


Pig.  17. 


angle  of  pressure  to  be  determined,  the  least 
number  of  teeth  Ijeing  N.  Suppose  the  gear  to 
engage   a   rack   having   standard  teeth,   then  will 

BC=-  =  -=-— .     D  will  be  the  last  point  of  con- 
P       N         N  ^ 

tact,  and  A  D  =  r,  the  radius  of  the  base  circle. 


A  C  =  A  B 


C  =  R-p 


2R_R(N-2) 


AC:AD::AD:AB,  hcnce,   A    D2  =  r2  =  A   BxAC  = 
R-^(N-2)  1  r,      /N  — 2 

_^^andr=Ry/-^. 

The  angle  of  pressure,  D  B  T,  is  equal  to  an- 
gle D  A  B  =  p  ,  and 


the  cos. 


P=R  = 


V^ 


N  — 2 
N 


=  jNzi2.      Hence    the 


N 


CDS.  of  the  angle  of  pressure  =1 


N-2 


N 


(5). 


By  substituting  in  the  above  fornnila,  it  will 
be  seen  that  for  a  12-tootlied  gear  to  engage  a 
rack  without  interference,  the  -angle  of  action 
must  be  24.1°,  and  for  15  teeth  the  angle  Avould 
be  '21  A°.    Again,  if  the   angle  be  15°,  the  least 


DEFECTS    OF    THE    INVOLUTE    SYSTEM.  35 

imiiiber  of  teetli  that  will  engage  -without  interference  will  be  30,  while  with  a  20°  angle  of 
pressure  the  least  number  would  be  18. 

46.  Defects  of  a  System  of  Involute  Gearing.  As  in  the  case  of  the  cycloidal  system,  it 
is  desirable  to  make  all  involute  gears  having  the  same  pitch  to  engage  correctly.  In  cycloidal 
gears  this  was  attained  1)}-  the  use  of  one  diameter  of  rolling  circle  for  all  geai-s  of  tlie  same 
pitch  (Art.  33,  page  20).  In  tlie  involute  system  we  assume  an  angle  of  oljliquity,  or  pressure, 
which  is  constant  for  all  geai-s ;  but  unless  this  angle  be  great,  gears  havings  so  few  as  12 
teeth  cannot  be  run  together  without  interference.  To  obviate  this  difficulty  we  must  adopt 
one  of  the  two  methods  already  described  (Art.  43,  page  32)  ;  namely,  the  undercutting  of  the 
interfering  flanks,  or  the  rounding  of  the  interfering  addenda.  Fii-st  consider  the  latter,  which 
is  illustrated  by  Plates  9  and  10.  We  have  seen  how  that  portion  of  the  gear  tooth  adden- 
dum lying  beyond  the  point  C  must  be  made  epicycloidal  in  order  to  engage  the  radial  i)art 
of  the  pinion  Hank  which  lies  within  the  base  circle;  also  that  the  j^inion  addenda  might 
be  wholly  involute  since  there  would  be  no  interference  with  the  gear  tooth  flank,  the  action 
between  the  latter  taking  place  without  the  base  circle.  But  if  a  12-toothed  gear  be  taken  as 
the  base  of  the  system,  it  will  be  necessary  to  round,  or  epicycloidally  extend  that  portion  of 
the  pinion  addendum  lying  beyond  the  point  K,  since  this  would  be  the  last  point  of  involute 
action  between  two  12-toothed  geai-s.  Therefore  when  the  12-tootlied  gear  engages  one  having 
a  greater  numl)er  of  teeth,  that  part  of  the  addendum  lying  beyond  this  point  will  no  longer 
engage  the  second  gear,  and  the  arc  of  contact  will  be  greatly  reduced.  Again,  suppose  a  pair 
of  30-toothed  geai-s  to  engage  (each  being  designed  to  engage  a  12-toothed  pinion),  the  only 
part  of  the  tooth  suitable  for  transmitting  a  uniform  motion  is  that  Iving  between  the  base 


36 


UNSYMMETEICAL    TEETH. 


circle  and  point  c,  Plate  9,  and  the  arc  of  contact  ^yould 
be  but  1.05  of  tlie  circular  pitch.  Now,  one  of  the  claims 
made  for  the  involute  tooth  is  that  the  distance  between 
the  centers  of  the  geare  nmj  be  changed  without  changing 
the  velocity  ratio ;  but  in  tliis  latter  case  it  cannot  be  done 
without  making  the  arc  of  contact  less  than  the  circular 
pitch. 

If  the  system  of  undercutting  the  flanks  be  adopted, 
the  addendum  will  be  wholly  involute ;  and  in  the  case  of 
Plate  9  all  of  the  pinion  addendum  would  have  been 
available  for  action,  but  the  pinion  flank,  within  the  base 
circle  would  have  been  cut  away  so  that  there  would  have 
been  no  action  of  the  gear  addendum  between  C  and  E. 
If,  however,  the  engagement  had  been  between  two  30- 
toothed  gears,  all  of  the  tooth  would  have  been  available 
for  action,  and  the  arc  of  contact  would  have  been  equal 
to  1.91  of  the  circular  pitch. 

Thus  it  will  be  seen  that  involute  gears  should  be  de- 
signed to  engage  the  geai-s  with  which  they  are  intended 
to  run,  if  the  best  results  would  be  attained.  This  would, 
of  coui-se,  prevent  the  use  of  the  ready-made  gear  or  cut- 
ter, but  would  insure  a  longer  arc  of  action  between  con- 
jugate teeth. 


UNSYMMETRICAL    TEETH.  87 

47.  Unsymmetrical  Teeth.  Fig.  18.  A  very  desirable,  although  little  used,  form  of  tooth 
is  that  knowu  as  the  unsymmetrical  tooth,  Avhich  usually  comltines  the  cycloidal  and  involute 
systems.  Fig.  18  illustrates  a  pinion  and  gear  liaving  the  same  numl)er  of  teeth  as  those  illus- 
trated by  Plate  4,  and  the  arc  of  contact  is  unchanged  ;  but  the  angle  of  pressure  is  much 
reduced,  and  the  strength  of  the  tooth  increased.  As  the  involute  face  of  the  tooth  is  designed 
to  act  only  when  it  may  be  necessary  to  reverse  the  geai-s,  and  when  less  force  would  usually  l)e 
transmitted,  the  angle  of  pressure  may  be  made  greater  than  ordinary.  In  this  case  the  angle 
is  24.1°,  which  is  sufficient  to  avoid  interference  in  a  standard  12-toothed  gear  (Art.  45, 
page  33).  But  this  angle  is  no  greater  than  the  maximum  angle  of  pressure  in  Plate  4. 
This  reinforcement  of  the  Ijack  of  the  tooth  makes  it  possible  to  use  a  much  greater  diameter 
of  rolling  circle ;  and  in  the  case  illustrated,  the  diameter  is  one-third  greater  than  the  radius 
of  the  pitch  circle.  This  increase  in  the  diameter  of  the  rolling  circle  would  have  lengthened 
the  arc  of  contact,  had  not  the  height  of  the  tooth  been  reduced  to  maintain  the  same  arc  as 
that  of  Plate  4. 

The  cycloidal  action  begins  at  C  and  ends  at  H ,  making  a  maximum  angle  of  pressure  of 
17°.  The  same  rolling  circle  has  been  used  for  the  face  and  flank  of  each  gear;  but  the  one 
rolling  within  the  pitch  circle  of  the  gear  might  have  been  much  increased  without  materially 
weakening  th.e  gear  tooth. 

The  hivolute  action  would  begin  at  D  and  end  at  B,  making  an  arc  of  contact  a  little  greater 
than  the  pitch. 


38  ANNULAR    GEARING. 

CHAPTER   V. 

ANNULAR     GEARING. 

48.  Cycloidal  System  of  Annular  Gearing.  If  the  center  of  the  pinion  lies  within  the  pitch 
circle  of  the  gear,  the  hitter  is  called  an  iitternal,  or  annular  gear.  The  solution  of  problems 
relating  to  tliis  form  of  gearing  diffei-s  in  no  wise  from  that  of  the  ordinar}-  external  spur  gear, 
save  in  the  consideration  of  certain  limitations  which  will  be  treated  of. 

49.  Limiting  Case.  Plate  11  illustrates  a  pinion  engaging  an  internal  and  an  external 
spur  gear.  The  pinion  has  6  teeth,  and  the  gears  have  13  teeth.  The  arc  of  contact  is  made 
equal  to  the  circular  pitch,  and  equally  divided  between  recess  and  approach.  The  pinion  has 
radial  flanlcs,  which  therefore  determines  the  diameter  of  the  describing  circle  for  the  addenda 
of  the  geare.  The  second  describing  circle.  2.  is  governed  by  conditions  which  will  appear  later. 
It  will  be  observed  that  the  addenda  of  the  auiuilar  gear  teeth  lie  within,  and  the  dedenda 
without,  the  pitch  circle.  The  height  of  the  teeth  is  governed  by  the  arcs  of  approach  and 
recess ;  and  the  construction  of  the  teeth  does  not  differ  from  the  limiting  case  considered  in 
Art.  26,  page  14,  and  Plate  5.  The  action  between  the  pinion  and  annular  gear  begins  at 
B.  and  ends  at  C.  the  pinion  driving. 

50.  Secondary  Action  in  Annular  Gearing.  We  have  already  seen.  Art.  10,  page  6,  that 
every  epicycloid  may  be  generated  by  either  of  two  rolling  circles,  which  differ  in  diameter  by 


SECONDARY    ACTION    IN    ANNULAR    GEARING.  39 

an  amount  equal  to  the  diameter  of  the  pitch  circle.  Also,  that  every  hypocycloid  may  he 
generated  hy  either  of  two  rolling  circles,  the  sum  of  the  diametei-s  of  which  sliall  equal  that 
of  the  pitch  circle  within  which  they  roll.  Tlius  the  addendum,  C  E,  of  the  pinion,  Plate  11, 
may  be  described  by  the  circle  2,  or  tlie  intermediate  circle  3.  But  in  this  case  the  circles  1 
and  2  are  so  chosen  that  the  intermediate  circle  3  is  the  second  describing  circle  for  the  hypo- 
cycloid  F  G ,  as  well  as  for  the  epicycloid  C  E ;  consecjuently  C  E  and  F  G  will  produce  a  uniform 
velocity  ratio,  the  contact  taking  place  from  A  to  D.  Tlie  addendum  C  E  has  contact  also  witli 
the  dedendum  C  F  along  the  path  A  C ;  hence,  during  a  part  of  the  arc  of  recess  there  nuist  be 
two  points  of  each  tooth  in  contact  at  the  same  time. 

The  plate  illustrates  the  contact  along  the  path  A  C  as  just  completed  ;  bnt  a  second  point 
of  contact  will  be  seen  on  circle  3,  between  F  and  E,  and  action  along  this  path  will  be  con- 
tinued to  D.  The  case  is  therefore  no  longer  a  limiting  one,  inasmuch  as  the  arc  of  contact  is 
greater  than  the  circular  pitch.  The  additional  contact  takes  place  during  the  arc  of  recess, 
which  is  also  advantageous. 

In  order  to  obtain  this  secondary  action,  the  sum  of  the  radii  of  the  inner  n)nl  outer  roUine/ 
circles  must  equal  the  distance  between  the  centers  of  pinion  and  gear.* 

For,  letting  r, ,  r.,,  and  r;,  l)e  the  radii  of  the  inner,  outer,  and  intermediate  rolling  circles, 
and  Rp,  Rg,  the  I'adii  of  jjinion  and  gear,  1-3  +  ri  =  Rg,  (G),  and  r.j  —  r.  =  Rp,  (7),  AuT.  10,  page  (3. 
Subtracting  the  second  equation  from  the  lii-st,  ri  +  r^  =  Rg  —  Rp  =  C  =  center  distance  (8). 

Platp:  12,  Fig.  1,  illustrates  the  same  pinion  and  gear,  the  teeth  having  been  described  by 
the  intermediate  circle  only.      In  this  case  the  action  takes  place  wholly  during  recess,  the  arc 

*  The  slntU'iit  is  roftMTod  to  Prof.  MacCord's  "  Kinomatii's,"  pages  104  to  10!)  inclusive,  for  a  very  complete 
demonstration  of  tliis  law,  together  with  other  limitations  of  annular  gears. 


40  LIMITATION    OF    INTERMEDIATE    DESCRIBING    CURVE. 

of  recess  being  the  same  as  before,  about  1^  times  the  circular  pitch.  Had  the  outer  describing 
circle  been  used  to  describe  the  dedenda  of  the  gear  teeth,  as  in  the  preceding  cases,  a  secondary 
action  would  have  taken  place  during  the  recess. 

Special  notice  should  be  taken  of  the  reduced  angle  of  pressure  in  the  secondary  action  of 
annular  gearing,  and  of  the  possibility  of  obtaining  a  great  arc  of  recess  with  little  or  no 
approaching  action.  These  advantages  are  very  apparent  in  Plate  11,  in  which  the  pinion 
engages  an  external  and  an  internal  gear  having  an  equal  number  of  teeth. 

51.  Limitations  of  the  Intermediate  Describing  Circle.  Plate  12,  Fig.  2.  Suppose  the 
inner  describing  circle,  1,  Plate  11,  to  be  increased  until  it  equals  the  diameter  of  the  pinion 
pitch  circle,  9f,  the  radius  of  the  intermediate  describing  circle  will  then  equal  the  center  dis- 
tance, 5]^,  and  the  outer  describing  circle,  2,  would  be  but  \'^  radius.  For  by  substituting 
Rp  for  rj  in  equations  6  and  8,  Art.  50,  we  shall  obtain  rg  =  Rg  —  Rp  =  C,  and  ro  =  C  —  Rp .  Plate 
12,  Fig.  2,  illustrates  this  case,  the  outer  describing  circle  not  being  employed. 

Since  the  pinion  pitch  circle  has  now  become  a  describing  curve,  there  will  be  an  approach- 
ing action ;  but  only  one  point  of  the  pinion  tooth  will  act,  as  the  diameter  of  the  describing 
circle  and  pitch  circle  being  equal  reduces  the  pinion  flank  to  a  point.  But  if  any  further 
increase  be  made  in  the  diameter  of  the  inner  circle,  which  is  equivalent  to  a  decrease  in  the 
intermediate  describing  curve,  an  interference  will  take  place  during  approaching  action ;  since 
the  curves  of  gear  and  pinion  teeth,  generated  by  a  circle  greater  than  the  pinion  diameter,  will 
cross  one  another,  which  would  make  action  impossible.  Hence,  the  7'adius  of  the  intet'mediate 
describing  circle  cannot  he  less  than  the  line  of  centers. 

52.  Limitations  of   Exterior  and  Interior  Describing  Circles.     Plate  12,  Fig.  3,     From 


LIMITATION    OF    EXTERIOR    AND    INTERIOR    DESCRIBING    CURVES.  41 

Art.  50,  page  39,  it  was  seen  that  the  sum  of  the  radii  of  tlie  exterior  and  interior  describing 
circles  must  equal  the  center  distance  if  a  secondary  action  be  obtained.  If  either  circle  be 
decreased  without  decreasing  the  other,  the  secondary  action  ceases ;  but  if  either  circle  be 
increased  without  an  equal  decrease  in  the  other,  thus  making  the  sum  of  their  radii  greater 
than  the  center  distance,  the  addenda  will  interfere.  Thus,  in  Plate  11,  a  decrease  in  descril>- 
ing  circle  2  would  produce  a  more  rounding  face,  and  C  E  would  fail  to  engage  F  G ;  but  had 
this  describing  circle  been  increased  in  diameter  without  a  corresponding  decrease  in  1,  C  E 
would  have  interfered  with  F  G.  Hence,  the  limit  of  the  sum  of  the  radii  of  the  exterior  and 
interior  deserihitiy  circles  is  the  center  distance. 

Plate  12,  Fig.  3,  illustrates  a  special  case  of  the  above  condition,  the  interior  describing 
circle  being  reduced  to  zero,  and  the  radius  of  the  exterior  circle  made  equal  to  the  center  dis- 
tance, thus  making  the  intermediate  describing  circle  equal  to  the  pitch  circle  of  the  gear. 
There  will  be  dou])le  contact  during  a  portion  of  the  arc  of  recess,  the  contact  beginning  at  A, 
and  following  the  outer  describing  circle  to  C,  and  tlie  intermediate  (or  in  this  case  the  pitch 
circle  of  the  gear)  to  D .  This  design  is  objectionable  in  that  the  secondary  action  takes  place 
with  only  one  point  of  the  gear  tooth. 

53.  The  Limiting  Values  of  the  Exterior,  Interior,  and  Intermediate  Describing  Circles  for 
Secondary  Action.  Since  ro-|-ri=C,  either  radius  will  equal  C,  when  the  other  becomes  zero; 
but  if  there  be  a  secondary  action,  the  mininuun  value  of  r^  may  not  be  zero,  for  r^  will  be  a 
maximum  when  r.,  is  a  minimum,  as  rj  +  rj  =  Rg,  rs  is  a  mininuim  when  ecjual  to  C  (Art.  52), 
and  substituting  this  value  in  the  last  equation,  r^  =  Rg  —  C .  Again  substituting  this  value  in 
the  equation,  r.,  +  rj  =  C ,  r.^  =  C  —  (Rg  —  C)  =  2  C  —  Rg. 


42  LIMITING    VALUES    OF    DESCTJBIXG    CIRCLES    FOE    SECONDARY    ACTION. 

Summarv  of  the  above  limiting  values  and  conditions  governing  secondary  action:  — 

ri  maximum  =  Rg  —  C  ;  r^  minimum  =  0  :  rg  +  r^  =  Rg .        (6) 

r^  maximum  =  C  ;  r.,  minimum  =  2  C  -  Rg ;  rs  —  r^  =  Rp .        (7) 

rg  maximum  =  Rg ;  rg  minimum  =  C;  Rg  —  Rp  =  C.      (8) 

54,  Practical  Case.  If  annular  geai-s  be  made  interchangeable  with  spur  geare,  it  will  be 
necessarv  to  have  the  number  of  teeth  in  the  engaging  gears  differ  by  a  certain  number  which 
will  depend  on  the  base  of  the  system.  This  is  due  to  the  limitation  in  the  sum  of  the  radii  of 
the  describing  circles,  Art.  52,  page  40.  Thus,  let  12  be  the  base  of  the  system,  and  it  is 
required  to  find  the  least  number  of  teeth  in  the  annular  gear  that  will  engage  the  pinion.  If 
the  pitch  l)e  2,  the  diameter  of  the  pinion  will  be  6,  and  that  of  the  describing  circles  3.  But 
since  the  center  distance  cannot  be  greater  than  the  sum  of  the  radii  of  the  describing  circles 
(in  this  case  3),  the  diameter  of  the  annular  gear  must  be  12,  and  the  least  number  of  teeth  in 
the  annular  gear  will  be  24. 

Using  the  notation  of  Plate  11.  and  Art.  17.  page  11,  N  l)eing  the  least  number  of  teeth 
in  the  gear,  and  n  the  least  number  in  the  pinion,  or  the  base  of  the  system:  — 

ni~r,„  Nrii  nNn  ^., 

C  =  2r,=  — ,  also  C=Rg-Rp  =  — -  — ,  hence  ^  =  ^-^.  or  2n  =  N. 
Tlie  least  number  of  teeth  in  the  annuJar  gear  n'iU  be  twice  that  of  the  base  of  the  si/stem. 

55.  Summary  of  Limitations  and  Practical  Considerations.  (</)  The  diameter  of  the  inter- 
mediate describing  circle  is  equal  to  the  diameter  of  the  pinion,  plus  the  diameter  of  exterior 
describing  circle,  or  diameter  of  gear  minus  interior  describing  circle.     (Art.  10,  page  6.) 


SUMMARY    OF    LIMITATIONS    AND    PRACTICAL    CONSIDERATIONS.  43 

(6)  There  will  be  .secondiiiy  action  only  when  the  sum  of  the  radii  of  the  exterior  and 
interior  describing  circles  is  equal  to  the  line  of  centers.     (Art.  50,  page  38.) 

(c)  'J'hc  radius  of  the  intermediate  describing  circle  cannot  l)e  less  than  the  cent^^r  distance. 
(Art.  51,  page  40.) 

(tZ)  The  sum  of  the  radii  of  exterior  and  interior  describing  circles  cannot  be  greater  than 
the  center  distance.     (Art.  52,  page  40.) 

(ti)  The  number  of  teeth  in  any  pair  of  gears  of  an  interchangeable  system  must  differ  by 
an  amount  equal  to  tlie  base  of  the  system.     (Art.  54,  page  42.) 

(/)  If  the  pinion  drives,  the  exterior  describing  circle  shouhl  be  the  greater  in  order  that 
the  arc  of  contact  may  be  chiefly  one  of  recess. 

(^)  If  the  gear  drives,  the  interior  describing  circle  should  be  the  greater,  and  the  pinion 
teeth  may  have  flanks  onl}-,  but  in  this  case  the  teeth  should  be  extended  slightly  beyond  the 
pitch  circle  in  order  to  protect  the  last  point  of  contact,  which  will  Ije  on  the  pitch  circle. 

56.  Involute  System  of  Annular  Gearing.  Fig.  19.  The  method  of  drawing  the  tooth 
outlines  for  the  involute  annular  gear  does  not  dift'er  from  that  of  the  spur  gear.  Pitch  lines 
liaving  been  determined,  the  base  circles  are  diawn  tangent  to  the  line  of  action,  and  the  invo- 
lutes of  those  base  circles  will  be  the  required  curves.  Care  must  be  used  in  obtaining  the 
length  of  the  teeth,  in  order  to  avoid  a  second  engagement  after  the  full  action  shall  have 
taken  place.  To  determine  if  this  interference  takes  place,  it  is  necessary  to  construct  the 
epitrochoid  of  the  point  of  the  pinion  tooth,  or  determine  the  path  of  least  clearance,  as  in 
Art.  28,  page  15. 


44 


INVOLUTE    SYSTEM    OF    ANNULAR    GEARING. 


Fig.  19  illustrates  an  annular  gear  of  20  teeth  en- 
gaging a  pinion  of  10  teeth,  the  angle  of  pressure 
being  20°.  The  pinion  driving  in  the  direction  indi- 
cated will  establish  the  first  i)oint  of  contact  at  A,  and 
the  last  point,  B,  ^^•ill  be  limited  by  the  height  of  the 

tooth,  in  this  case  - .     The  limit  of  the  gear  tooth  will 

be  determined  by  the  arc  drawn  from  the  center  of 
gear  through  the  point  A.  Any  extension  of  the  in- 
volute beyond  this  point  will  interfere  with  the  pinion 
flank.  The  stronger  form  of  the  annular  gear  tooth 
permits  of  a  greater  clearance,  which  it  is  advantageous 
to  adopt. 

If  the  pinion  and  gear  differ  but  little  in  diameter, 
it  is  desirable  to  use  the  cycloidal  system,  in  which  case 
the  interference  may  be  more  easily  avoided.  It  sliould 
also  be  noted  that  the  advantages  to  be  derived  from 
an  increase  in  the  arc  of  contact  and  a  decrease  in  the 
angle  of  pressure  are  to  be  obtained  only  by  the  use  of 
the  latter  system. 


THEORY  OF  BEVEL  GEARING.  45 

CHAPTER   VI. 

BEVEL     GEARING. 

57.  Theory  of  Bevel  Gearing.  In  the  cases  previously  considered,  the  elements  of  the  teeth 
were  parallel,  the  surfaces  having  been  generated  by  a  right  line  which  was  either  an  element 
of  a  rolling  cylinder,  as  in  the  cycloidal  system,  or  by  an  element  of  a  flexible  band  parallel  to 
the  axis  of  a  cylinder  from  which  it  was  unwrapped,  as  in  the  involute  system.  All  sections 
of  the  teeth  made  by  planes  perpendicular  to  the  axis  were  alike,  and  therefore  it  was  only 
necessary  to  consider  one.  Under  these  conditions  the  pitch  cyclinder  became  a  pitch  circle, 
and  the  describing  cylinder  a  describing  circle.  If  we  now  consider  the  axes  of  the  gears  as 
intersecting,  the  friction  cylinders  will  become  friction  cones,  the  describing  cylinder  will  be  a 
describing  cone,  and  the  elements  of  the  teeth  will  converge  to  the  point  of  intersection  of  the 
axes,  making  all  sections  of  the  teeth  to  differ  from  one  another. 

Fig.  20,  page  46,  illustrates  this  case.  A  C  B  and  BCD  are  two  friction  cones,  or  pitch  cones, 
having  axes  G  C  and  H  C.  The  outlines  of  the  teeth  are  drawn  on  the  spherical  base  of  the 
cone,  that  portion  of  the  curve  lying  outside  the  pitch  cone  being  a  spherical  epicycloid,  and 
that  within,  a  spherical  h>^ocycloid.  The  dedendum,  or  surface  of  the  tooth  lying  within  the 
pitch  cone  A  C  B,  was  described  by  the  element  E  F  C  of  the  describing  cone,  which  is  shown  as 
generating  the  addendum  of  the  pinion  tooth.  Only  that  portion  of  the  surface  described  by  E  F 
would  be  used  for  the  pinion  tooth,  the  length  of  the  gear  tooth  having  been  limited  as  shown. 
The  describing  cone  employed  for  generating  the  addendum  of  gear,  and  dedendum  of  pinion, 


46 


CHARACTER  OF  CURYES  IN  BEYEL  GEARING. 


is  not  shown;  but  the  diameter  of 
its  base  would  be  governed  by  laws 
siniilar  to  those  already  considered 
for  limiting  the  diameters  of  rolling 
circles,  Art.  32,  page  18. 

58.  Character  of  Curves  employed 
in  Bevel  Gearing,  The  cycloidal 
BEYEL  TOOTH  has  already  been  con- 
sidered in  the  previous  article,  and 
the  curve  does  not  differ  from  that 
employed  in  spur  gearing,  save  that 
it  is  described  on  the  surface  of  a 
sphere. 

It  is  important  to  note  that  no 
tooth  can  be  made  with  a  radial  flank, 
since  no  circular  cone  can  be  made 
to  generate  a  plane  surface  by  roll- 
ing Avithin  another  cone,  but  the 
flank  may  approximate  closely  to 
such  plane. 

The    INYOLUTE    BEYEL    TOOTH  is 

one  ha^dnof  a  grreat  circle  for  its  Hne 


Fig.  20 


TREDGOLD    APPROXIMATION. 


47 


of  action.  Fig.  21  illustrates  a  crown  gear  of  this  type.  A  C 
is  a  great  circle  of  the  sphere  A  D  C  E,  and  is  tangent  to  the 
circles  A  E  and  DC.  If  the  circle  A  C  be  rolled  on  D  C,  so  as 
to  continue  tangent  to  D  C  and  A  E,  the  point  B  will  describe 
the  spherical  involute  G  B  F.  Conjugate  teeth  described  by 
this  process  maintain  their  velocity  ratio  constant,  even  while 
undergonig  a  slight  cliange  in  their  shaft  angles,  tlius  conform- 
ing to  the  general  character  of  involute  curves. 

The  OCTOID  BEVEL  TOOTH  is  One  having  a  plane  surface 
for  the  addendum  and  dedendum,  the  plane  being  such  as 
would  cut  a  great  circle  from  the  surface  of  the  sphere.  In 
Fig.  22,  G  F  is  the  plane  Avhich  cuts  the  surface  of  the  tooth 
shown  at  B .  The  line  of  action,  from  which  the  tooth  takes 
its  name,  is  indicated  l)y  the  curve  B  C  E  B  H  K  .  This  tooth 
was  the  invention  of  Hugo  Bilgram,  and  is  frequently  confused 
with  the  involute  tooth.  It  can  be  formed  in  a  practical  manner 
by  the  molding-planing  process.  The  IJilgram  machine,  de- 
signed to  plane  this  toolli,  is  described  m  tlie  Journal  of  the 
Franklin  Institute  for  August,  188G,  and  in  the  American  Ma- 
chinist for  ]\Iay  9,  1885. 

59.  Tredgold  Approximation.  Because  of  the  dilficulty  in- 
volved in  describing  the  tooth  form  on  the  surface  of  a  sphere, 


Fig.  21. 


DRAFTING  THE  BETEL  GEAR. 

it  is  custoniar)-  to  draw  the  outline  on  the  developed  surface  of 
a  cone    which   is   tangent   to   the   sphere    at   the   pitch   circle. 
Tliis  cone  is  called  the  normal,  or  back  cone.     Plate  13  il- 
lustrates a  sphere  A  B  D,  from  which  the  pitch  cones  A  C  B 
and  BCD  have  been  cut.      Tangent  to  the  sphere  at  the 
pitch  circles.   A  B  and  B  D.   are   the  normal  cones  A  G  B 
and  B  H  D.  the  elements  of  which  are  perpendicular  to 
the  intersecting  elements  of  the  pitch  cones.      The 
error  in  the  tooth  curve  due  to  this  approximation 
is  so  small  as  to  be  inappreciable,  save  in  exag- 
gerated  cases :   and  the   method   is   always  em- 
ployed for  the  drafting  of  bevel  geai"S. 


60.    Drafting  the  Bevel  Gear.     Plate 
13,  and   Fig.   23.      The   drawing    usually- 
required  is  that  illustrated  by  Fig.  23, 
which  is  a  section  of  a  gear  and  pin- 
ion, together  with  the    development  of 
a   portion  of    the  outer  and  inner  nor- 
mal cones,  only  the  tooth  curves  being 
^o^   omitted. 

^^  The  names  of  the  parts  of   a  bevel 

Q      gear  are   also  given,   and   the  lettering 


Fig.  23. 


DRAFTING  THE  BETEL  GEAR.  49 

corresponds  to  that  of  Plate  13,  wliieli  latter  will  he  used  to  illustrate  the  method  of 
drawing. 

A  B  and  B  D,  Platk  13,  aie  the  i)itch  diameters  of  a  gear  and  pinion  with  axes  at  90",  and 
having  15  and  lii  teeth  respectively,  the  pitch  being  3,  when  drawn  to  the  scale  indicated. 
The  pitch  diameters  being  5"  and  4",  lay  off  C  K  on  the  center  line  of  gear,  equal  to  one-half 
the  pitch  diameter  of  pinion,  and  C  L  on  the  center  line  of  pinion,  equal  to  one-half  the  pitch 
diameter  of  tlie  gear.  Through  these  points  draw  the  pitch  lines  perpendicular  to  the  axes  of 
the  gears,  and  in  this  case  peipendicular  to  each  other.  Draw  the  pitch  cones  A  C  B  and  BCD, 
and  perpendicular  to  these  elements  draw  G  A,  G  B  H,  and  H  D,  elements  of  the  normal  cones. 
Having  figured  the  addendum  and  dedendum  of  the  teeth,  lay  off  on  the  normal  cone  of  pinion 
B  M  and  B  N ,  D  0  and  D  Q ,  and  from  these  points  draw  lines  converging  to  the  apex  of  the  pitch 
cones.  Similarly  lay  off  addenda  and  dedenda  of  gear,  limiting  the  length  of  the  face  at  R  by 
drawing  the  elements  of  the  inner  normal  cones  at  R  S  and  R  T.  The  face  B  R  should  not  be 
greater  than  one-third  B  C,  by  reason  of  the  objectionable  reduction  in  small  end  of  teeth. 
Complete  the  gear  blank,  or  outline,  by  drawing  the  lines  limiting  the  thickness  of  the  gear, 
diameter  and  length  of  hub,  diameter  of  shaft,  etc.,  details  which  are  matters  of  design. 

The  development  of  the  normal  cone  of  the  gear,  B  G  A,  will  be  a  circular  segment  described 
with  radius  G  B,  and  equal  in  length  to  the  circumference  of  the  pitch  circle  of  the  gear. 
Since  there  are  15  teeth  in  the  gear,  the  developed  pitch  circle  will  be  divided  into  15  parts, 
as  shown,  and  the  circular  pitch  be  thus  determined.  But  it  is  unnecessary  to  obtain  the 
complete  development  as  shown  in  the  plate,  since  the  shape  of  one  tooth  and  space  is  alone 
required.  Therefore,  space  off  on  a  portion  of  tlie  arc  of  the  developed  pitch  circle,  the  cir- 
cular pitch,  B  V,  whicli  is  equal  to  -.     Draw  the  addendum  and  dedendum  circles  with  radii 


60  DRAFTING  THE  BEVEL  GEAR. 

equal  to  distance  of  these  circles  from  the  apex  of  the  normal  cone,  which  in  the  case  of  the 
gear  will  be  G  E  and  G  F . 

Next  determine  the  tooth  curve  as  for  spur  gears,  using  the  developed  pitcli  circle  instead 
of  the  real  pitch  circle.  In  the  case  illustrated,  the  curve  is  involute.  B  W  is  a  part  of  the 
line  of  action,  making  an  angle  of  75°  with  G  H,  the  line  of  centers.  The  base  circles  drawn 
tangent  to  this  line  will  be  the  circles  from  which  the  involutes  are  described.  Had  the  cycloidal 
system  been  employed,  the  diameter  of  the  rolling  circle  would  have  been  made  dependent  on 
the  diameter  of  the  developed  pitch  circle,  instead  of  the  pitch  diameter  A  B . 

In  like  manner  obtain  the  development  of  the  inner  normal  cones,  having  S  R  and  T  R  for 
elements,  and  describe  the  true  curves  of  the  small  end  of  teeth.  These  pitch  circles  may  be 
drawn  concentric  with  the  developed  pitch  circles  of  the  outer  cones,  or  with  S  and  T  as  centers, 
the  latter  being  the  method  commonly  adopted.  Both  methods  have  been  employed  in  the 
plate.  If  the  development  of  the  inner  pitch  cone  of  gear  be  drawn  from  the  center  G,  the 
reduced  pitch,  and  thickness  of  tooth,  may  be  obtained  by  drawing  the  radial  lines  from  the 
development  of  the  outer  cone  as  shown  by  the  fine  dotted  lines.  The  addendum  and  deden- 
dum  circles  will  be  described  with  radii  S  Z  and  S  Y,  and  the  tooth  curves  may  be  drawn  by 
determining  the  reduced  rolling  circle,  if  the  gear  be  cycloidal,  or  the  reduced  base  circle  if  the 
involute  system  be  employed. 

A  second  method  for  describing  the  teeth  on  the  inner  normal  cone  would  be  to  base  it  directly 
on  the  reduced  pitch,  which  may  l)e  determined  by  dividing  the  number  of  teeth  by  the  diameter 
of  the  base  of  the  pitch  cone  at  tliis  point.     In  the  plate,  the  value  of  P  for  small  end  of  teeth 

•       15  12 

is  T-TT  for  gear,  or  — r  for  pinion  =  4.6  =  P.  The  addendum,  dedendum,  circular  pitch,  etc.,  may 
now  be  obtained  from  this  value  of  P,  as  was  done  in  the  case  of  the  outer  pitcli  cone.     In  like 


FIGURING    BEVEL    GEARS. 


51 


manner  we  may  o])tain  any  other  section 
of  the  tooth,  although  a  third  section  is 
seldom  required. 

6i.  Figuring  the  Bevel  Gear  with  Axes 
at  90°.  Figs.  24  and  25.  The  dimensions 
required  for  the  figuring  of  a  pair  of  bevel 
gears  will  be  :  — 

First :  Those  re([uired  for  general  refer- 
ence, and  consisting  of  pitch  diametei-s, 
number  of  teeth  (or  pitch),  face  (K),  thick- 
ness of  gears  (L  and  M)  (U  and  V),  diameter 
and  length  of  hubs. 

Second :  In  addition  to  the  above,  the 
pattern  maker  and  machinist  will  require, 
for  the  turning  of  the  blank,  the  outside 
diameter,  backing,  angle  of  edge,  angle  of 
face. 

Third :  The  cutting  angle  will  be  re- 
quired for  cutting  the  teeth. 

The  figures  required  for  the  first  set  of 
dimensions  are  all  matters  of  design,  but 
the  second  and  third  dimensions  must  be 


.H4-L-t£. 


I      BACKING 

Tr — 


Wm^^ 


% 


52 


BEVEL    GEARING. 


determined  from  the  data  given  in  the  first.  To  obtain  these  it  is  necessar}^  to  figure  the  five 
dimensions  indicated  in  Fig.  25,  three  of  which,  A,  B,  and  C,  are  angles,  and  two,  E  and  F,  are 
necessary  to  determine  the  outside  diameter  and  backing.  Only  one  of  these,  A,  is  used 
directly.  B  is  called  the  angle  increment,  C  the  angle  decrement,  E  is  one-half  the  diameter 
increment  of  the  pinion,  and  F  is  equal  to  one-half  the  diameter  increment  of  the  gear. 
In  the  similar  right  triangles  a  b  t  and  t  r  m,  Fig  25, 


tan   A 


t  a  b  =  m  t  r 


tan   B 


a  b  = 


D' 


--f 


2  sin   A         2  sin  A 


d' 


P  d' 


t   m 


cos  A 


F  =  -  sin  A 


2  sin  A 

The  angle  decrement,  C,  is  sometimes  made  equal  to  B,  in  which  case  the 
dedendum  of  the  tooth  at  the  small  end  will  be  greater,  as  shown  by  the  line 
h  k ;   but  if  the  bottom  line  of  the  tooth  be  made  to  converge  to  the  apex  of 
the  pitch  cones,  the  angle  t  a  h,  or  C,  will  be  determined  as  follows: 

h  t  8  P         9     ^     sin   A  _   2.25  sin  A 

t  a  d'  4  n  n  ' 

2  sin  A 

Having  determined  these  values,  it  is  only  necessary  to  combine  them  with  those  fixed  by 
the  design  to  complete  the  figuring  of  the  gear  as  shown  in  Fig.  24. 

The  angles  should  be  expressed  in  degrees  and  tenths,  rather  than  in  degrees  and  minutes. 
It  is  also  of  importance  that  the  outside  diameter  and  backing  be  figured  in  decimals,  to  thou- 
sandths, rather  than  in  fractional  equivalents. 


tan  C 


Fig.  25. 


BEVEL    GEAR    TABLE.  53 

62.  Bevel  Gear  Table  for  Shafts  at  90".  In  order  to  facilitate  the  figuring  of  bevel  geai-s, 
tables  or  charts  of  the  principal  values  are  commonly  employed.  Such  charts  also  make  the 
figuring  possible  to  those  unfamiliar  with  the  solution  of  a  right  triangle.  Some  are  designed 
to  solve  the  problems  graphically,  while  others,  like  the  following,  pages  54  and  55,  consist  of 
the  trigonometrical  functions  for  geare  of  the  proportions  commonly  employed. 

Description  of  Table.* 

Column  1.    Ratio  of  Pinion  to  Gear.  ,, 

Column  2.    Ratio  of  Pinion  to  Gear  expressed  in  decimals,  or  tang  of  center  angle.      ^^"  *  ~  jy  "=  ju  ' 

Column  3.    Center  angle  of  Pinion  corresponding  to  tangent  in  column  2. 

Column  4.  Ten  times  the  angle  increment  lor  a  Pinion  of  10  teeth.  This  increased  value  is  employed  to 
simplify  the  figuring  of  gears  having  other  tlian  10  teeth.  Thus,  the  angle  increment  for  miter  geare 
(1  to  1)  having  10  teeth  would  be  8.2°,  and  for  14  teeth,  \j  of  this  value  or  J|.  There  is,  of  course,  a 
slight  error  in  deriving  the  angle  increment  for  any  number  of  teeth  from  these  values,  in  that  the 
tangent  and  arc  do  not  vary  alike,  but  the  error  is  inapjn-eciable  for  small  arcs. 

Column  5.    The  diameter  increment  for  a  Pinion  of  one  pitch,  hence  equal  to  2  cos  A. 

Column  6.    Center  angle  for  Gear,  or  90°  —  A. 

Column  7.  Ten  times  the  angle  increment  for  Gear  of  10  teeth,  which,  of  course  equab  that  of  the 
engaging  Pinion. 

Column  8.    Diameter  increment  for  a  Gear  of  one  pitch,  hence  equal  to  2  sin  A. 

Use  of  Table.  —  In  columns  1  or  2  find  the  value  corresponding  to  the  ratio  of  given 
gears.     Against  this  value,  in  3  and  6,  the  center  angles  for  pinion  and  gear  are  given. 

The  angle  increment  may  be  found  by  dividing  the  value  in  4  by  the  number  of  teeth  in 
tlie  pinion,  or  by  dividing  the  value  in  7  by  the  number  of  teeth  in  the  gear. 

The  diameter  increment  for  the  pinion  is  obtained  by  dividing  the  value  in  5  by  P ,  and  tliat 
for  the  gear  by  dividing  the  value  in  8  by  P. 

The  value  of  the  angle  C  may  be  determined  with  sufficient  accuracy  by  making  it  |  of  B . 

*  The  plan  of  this  table  is  that  adopted  by  Mr.  George  B.  Grant.  See  "American  Machinist/'  Oct.  31,  1885,  and 
"A  Treatise  ou  Gear  Wheels,"  page  90. 


54 


BEVEL    GEAR    TABLE. 


BEVEL   GEAR    TABLE  FOR   SHAFTS   AT   90°. 


d'+2E-; 

Fig.  26. 


PEOPORTION 

OF 

PINION 

TO  GEAR. 

PINION. 

GEAR. 

A. 

B. 

2  E. 

900  — A. 

B. 

2  F. 

Center 
Angle. 

Angle 

Increment. 

Divide 

by  n. 

Diameter 

Increment, 

Divide 

by  P. 

Center 
Angle. 

Angle 

Increment 

Divide 

by  N. 

Diameter 

Increment 

Divide 

by  P. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

1 

1 

1.000 

45. 

80.5 

1.414 

45. 

80.5 

1.414 

9 

10 

.900 

41.98 

76.0 

1.486 

48.02 

84.5 

1.337 

8 

9 

.888 

41.63 

75.6 

1.495 

48.37 

85.0 

1.329 

7 

8 

.875 

41.18 

75.0 

1.504 

48.82 

85.5 

1.317 

6 

7 

.857 

40.60 

74.1 

1.518 

49.40 

86.3 

1.302 

5 

6 

.833 

39.80 

73.0 

1.536 

50.20 

87.4 

1.280 

4 

5 

.800 

38.66 

71.1 

1.562 

51.34 

88.8 

1.249 

7 

9 

.777 

37.85 

70.0 

1.579 

52.15 

89.6 

1.228 

3 

4 

.750 

36.83 

68.5 

1.600 

53.17 

90.8 

1.200 

5 

( 

.714 

35.53 

66.2 

1.628 

54.47 

92.5 

1.162 

7 

10 

.700 

34.99 

65.1 

1.638 

55.01 

93.0 

1.147 

o 

3 

.666 

33.68 

63.2 

1.664 

56.32 

94.5 

1.109 

5 

8 

.625 

32.00 

60.4 

1.696 

58.00 

96.3 

1 .060 

3 

5 

.600 

30.96 

58.7 

1.715 

59.04 

97.3 

1.029 

^     4 

7 

.571 

29.75 

56.6 

1.736 

60.25 

98.5 

.992 

\lu     - 

9 

.555 

29.05 

55.4 

1.748 

60.95 

99.1 

.971 

-4^1 

2 

.500 

26.56 

51.0 

1.789 

63.44 

101.4 

.894 

-^     4 

9 

.444 

23.94 

46.3 

1.827 

66.06 

103.6 

.812 

BEVEL    GEAR    TABLE. 


55 


BEVEL   GEAR   TABLE   FOR   SHAFTS 

AT  90°. 

>ORTION 
OF 

NION 
GEAR. 

PINION. 

GEAR. 

PRO] 

A. 

B. 

2  E. 

900 -A. 

B. 

2  F. 

PI 
TO 

Center 
A  iigle. 

Angle 

Increment. 

Idvide 

Diameter 
Increment. 

Divide 

Center 
Angle. 

Angle 

Increment. 

Divide 

Diameter 

Increment. 

Divide 

by  n. 

by  P. 

by  N. 

by  P. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

3:    7 

.428 

23.20 

45. 

1.838 

66.80 

104.1 

.788 

2:    5 

.400 

21.80 

42.5 

1.857 

68.20 

105.2 

.743 

3:    8 

.375 

20.55 

40.3 

1.873 

69.45 

106.0 

.702 

1:    3 

.333 

18.43 

36.1 

1.897 

71.57 

107.4 

.632 

3:10 

.300 

16.70 

■32.8 

1.915 

73.30 

108.4 

.575 

2:    7 

.285 

15.95 

31.4 

1.923 

74.05 

108.8 

.549 

1:    4 

.250 

14.03 

27.8 

1.940 

75.97 

109.8 

.485 

2:    9 

.222 

12.53 

24.6 

1.952 

77.47 

110.5 

.434 

1:    5 

.200 

11.31 

22.5 

1.961 

78.69 

111.0 

.392 

2:11 

.181 

10.30 

20.4 

1.968 

79.70 

iii.a 

.357 

1:    6 

.166 

9.46 

18.8 

1.973 

80.54 

111.6 

.329 

2:13 

.153 

8.75 

17.4 

1.977 

81.25 

111.8 

.304 

1:    7 

.143 

8.13 

16.4 

1.980 

81.87 

112.0 

.283 

2:15 

.133 

7.60 

15.0 

1.982 

82.40 

112.1 

.265 

1:    8 

.125 

7.12 

14.2 

1.985 

82.88 

112.2 

.248 

2:17 

.117 

6.70 

13.3 

l.<)86 

83.30 

112.3 

.233 

1:    9 

.111 

6.33 

12.6 

1.988 

83.67 

112.4 

.221 

1:10 

.100 

5.70 

11.3 

1.990 

84.30 

112.6 

lioo 

P  =  -  =  "  ; 


P-  =  ^: 


.       d'        n 
tan  A  =  ^,  =  ^ 


tan  B  = 


tan  C  = 


2  sin  A 
n 


2.25  sin  A 


2  E  =  p  cos  A ; 


F  =  p  sin  A. 


56 


BEVEL    GEARS    WITH    AXES    AT    AXY    ANGLE. 


63.  Bevel  Gears  with  Axes  at  any  Angle. 
If  the  axes  of  the  gears  intersect  at  angles  other 
than  90°,  tlie  drawing  of  tlie  Wanks  and  devel- 
opment of  the  teeth  do  not  differ  from  the  cases 
already  described.  The  figuring  required  is  that 
indicated  in  Fig.  27,  those  in  the  heavy  face 
being  used  to  determine  the  other  values,  and 
not  appearing  on  the  finished  drawing. 

.        ./  sin  a 

tan  A   = ■ ; 


tan    A  = 

sin  a 

N 

-  +  cos  a 

!^ 

tan   B  = 

2  sin  A 
= or 

n 

tan   C  : 

2.25  sin  A 

+  COS  a 


2  sin  A^ 
N       ' 


E  =  —  cos  A  ; 
P 


—  sin  A  ; 
P 


2.25  sin  A' 
N 


E'  =  —  cos  A'. 
P 


F'  =  —  sin  A'. 
P 


rig.  27. 


Or,  the  values  for  E ,  F,  E',  and  F'  may  be .  ob- 
tained from  the  table  for  shafts  at  90°,  pages  54 
and  55  by  determining  the  center  angles  A  atid 
A',  and  finding  the  values  for  2  E  and  2  F,  corre- 
sponding to  each  gear  separately. 


ODONTOGRA.PHS.  57 


CHAPTER   VIT. 

SPECIAL    FORMS    OF    ODONTOIDS,    NOTATION,    FORMULAS.    ETC. 

64.  Odontographs  and  Odontograph  Tables.  If  tooth  curves  are  to  be  drawn  according  to 
some  established  system,  as  in  the  involute  when  the  angle  of  pressure  is  constant,  or  in  the 
cycloidal  when  but  one  diameter  of  rolling  circle  be  used,  it  may  be  desirable  to  employ  some  of 
the  approximate  methods  for  shortening  the  operation  of  describing  the  outline  of  the  teeth. 
While  it  is  unnecessary  for  the  student  to  familiarize  himself  with  the  theory,  or  even  the 
details,  of  operating  the  various  systems  for  approximating  these  curves,  it  is  essential  that  a 
knowledge  be  had  of  the  more  useful  tables  and  methods  to  which  reference  may  be  made 
when  required.  This  is  particularly  true  in  the  case  of  the  involute  tooth,  which  is  the  one 
most  used. 

Three  methods  are  employed  for  approximating  tlie  odontoidal  curves. 

First,  by  circular  arcs,  the  centers  and  radii  of  which  are  given  in  tables,  or  established  by 
instruments  designed  for  this  purpose. 

Second,  l)y  curved  templets  from  which  the  curves  may  be  traced  directly. 

Third,  by  ordinates. 


58 


ODONTOGRAPHS. 


_LINL_OF_FLANK 


GRANT'S   EPICYCLOIDAL   ODONTOGRAPH.i 


3  P.   21  T 


65.  The  Three  Point  Odontograph,  designed 
by  Mr.  George  B.  Grant,  is  a  table  for  the 
face  and  flank  radii,  and  location  of  centers 
for  circular  arcs  approximating  the  true  curves 
of  epicycloidal  teeth.  It  is  designed  for  that 
system  which  has  for  its  base  a  twelve-toothed 
gear  with  radial  flanks.     Art.  33,  page  20. 

In  using  this  odontograph,  proceed  as  fol- 
lows :  Draw  the  pitch  circle,  addendum,  and 
dedendum  circles,  and  space  the  pitcli  circle 
lor  the  teeth.  Obtain  the  radins  of  the  cir- 
cle for  the  flank  centers  by  laying  off  out- 
side the  pitch  circle  the  tabular  distance  for 
flanks,  as  given  in  the  sixth  column,  observing 
that    this    value    must    be    divided    by    the 


Tfetii  in 

Fo 
For  a 

R  ONE  Diametral  Pi 

rcii 

nE    BY 

NUMUF.R    01 

NY    OTHER 

Pitch  divi 

Ge 

All 

THAT 

PiTl'II 

Faces 

Flank 

Kxact 

Intervals 

Kad. 

Dis. 

Kad. 

Uis. 

12 

12 

2.01 

.06 

00 

00 

m 

13-U 

2.04 

.07 

15.10 

9.43 

lOi 

15-lG 

2.10 

.09 

7.80 

3.46 

1"' 

17-18 

2.14 

.11 

6.13 

2.20 

20 

19-21 

2.20 

.13 

5.12 

1..57 

23 

22-24 

2.2G 

.15 

4.50 

1.13 

27 

25-29 

2.33 

.16 

4.10 

.96 

3:3 

30-36 

2.40 

.19 

3.80 

.72 

42 

37-48 

2.48 

.22 

3.52 

.63 

58 

49-72 

2.60 

.25 

3.33 

.54 

97 

73-144 

2.83 

.28 

3.14 

.44 

290 

145-300 

2.92 

.31 

3.00 

.38 

00 

Rack 

2.96 

.34 

2.96 

.34 

1  By  permission  from  Grant's  "Treatise  on  Gear  Wheels." 


ODONTOGRAPHS. 


59 


GKAXT'S   INVOLUTE   ODONTOGKAPH.i 


diameter  pitch.  Similaiiy,  ubtaiii  the  circle 
for  the  face  centers,  which  will  be  drawn 
inside  the  pitch  circle.  Tlien,  with  the 
corresponding  radii  for  the  given  nunil)er  of 
teeth  (divided  by  the  pitch),  describe  the 
required  curves. 

66.  The  Grant  Involute  Odontograph  gives 
a  very  close  approximation  to  the  involute 
curve.  It  is  designed  for  the  system  of  15° 
angle  of  pressure  with  epicycloidal  extension. 
All  gear  teeth  made  by  this  odontograph  will 
engage  a  12-toothed  gear  without  interference. 

Having  drawn  the  pitch  circle,  addendum, 
and  dedendum  circles,  and  spaced  the  teeth, 


NfMHER 

OP  Teeth 


1-2 

1:5 
U 
1.-. 

10 
17 

18 
19 

20 
21 
22 
23 

24 

2.'> 
2(3 
27 

28 


Divide  iiv  the 
Diametral  1'it<h 


Face 
Radius 


2.51 
2.02 


2i)2 
3.02 
3.12 
3.22 

3.32 
3.41 
3.49 
3.57 

3.04 
3.71 
3.78 
3.8.-) 
3.92 


Flank 
Radius 


.96 
1.09 
1.22 
1.34 

1.40 
1..58 
1.09 
1.79 

1.89 
1.98 
2.06 
2.1.5 

2.24 
2.33 
2.42 
2.50 
2.59 


NiTMBER 

OF  Teetu 


29 
30 
31 
32 

33 
34 
35 
30 

37-40 
41-45 
46-51 
52-00 

61-70 

71-90 

91-120 

121-180 

181-360 


Divide  by  the 
Diametral  Pitch 


Face 
Radius 


3.99 
4.06 
4.13 
4.20 

4.27 
4.33 
4.39 
4.45 


Flank 
Radius 


2.67 
2.76 
2.85 
2.93 

3.01 
3.09 
3.16 
3.23 


4.20 
4.63 
5.06 
5.74 

6..o2 

7.72 

9.78 

13.38 

21.02 


>  By  permission  from  Grant's  "  Treatise  on  Gear  Wheels. 


60 


OD  ONTO  GRAPHS. 


obtain  the  circle  of  centers  for  face  and  flank  radii  by  describing  a  circle  tangent  to  the  15° 
line  of  pressure.  This  circle  will  be  one-sixtieth  of  the  pitch  diameter  inside  the  pitch  circle. 
Next,  obtain  the  face  and  flank  radii  from  the  table,  and,  having  divided  the  values  by  the 
diameter  pitch,  describe  the  required  arcs.  Observe  that  the  arc  for  the  flank  is  drawn  from 
the  pitch  circle  to  the  base  circle,  or  circle  of  centers,  the  remainder  of  the  flank  being  radial. 
The  method  of  drawing  the  rack  tooth  will  be  evident  from  the  figure. 


Fig.  28. 


-"tH^^^-, 


67.  Willis's  Odontograph.  Among  those  of  the 
first  tjpe,  the  oldest,  best  known,  and  least  accurate, 
are  the  odontographs  designed  by  Professor  Willis. 
When  used  for  gears  having  a  large  number  of  teeth, 
the  error  is  very  slight ;  but  in  the  case  of  involute  teeth 
of  small  number  it  is  very  noticeable.  Figure  28  illus- 
trates the  application  of  this  instrument  to  the  drawing 
of  curves  of  the  cycloidal  system.  The  centers  for  the 
circular  arcs,  designed  to  approximate  tlie  curves,  are 
/  \       \        i  found  on  the  straight  edge  AB,  and  at  a  distance  from 

'  \       A       I  i]^Q  2ero  point  of  the  scale  to  be  found  in  the  publi-shed 

table  accompanying  the  instrument. 

The  theory  and  application  of  these  odontographs 
is  clearly  treated  of  in  the  instructions  accompanying 
these  instruments,  also  in  Stahl  and  Wood's  "Elements 
of  Mechanism,"  pages  114-123,  and  more  briefly  in  MacCord's  "  Kinematics,"  pages  172-174. 


THE    ROBINSON    AND    KLEIN    ODONTOGKAPHS. 

68.  The  Robinson  Odontograph  differs  from 
the  preceding  in  that  it  is  an  instrument  hav- 
ing a  curved  edge  which  is  used  as  a  templet 
to  trace  the  tooth  curve,  tables  being  used  to 
determine  the  position  of  the  instrument  with 
relation  to  the  i:)itch  circle. 

Fig.   20  illustrates   the   instrument  in  })osi- 
tion.     The  curve   B  C  A  is  a  logarithmic  spiral, 
and  the  curve  B  F  H  the  evolute  of  the  fii-st,  and 
therefore  a  similar  and  equal  spiral.     By  means 
of  this  instrument,  in  connection  with  the  pul>- 
lished  tables   accompanying   it,    involute   teeth  may  be  drawn 
as  well   as  cycloidal,  and   a  much  larger  range  of   the  latter 
is  i)ossible  than   is  afforded  by  the  Willis  odontograph.     The 
theory  of    tliis   instrument  is   best   treated    by  Professor  Rol> 
inson  in  Xan  Nosti-and's  Edectie  Magazine  for  July,  1876,  and 
Van   Nostrand's    "Science    Series,"   No.   24.      Also  see  Stahl 
and  Wood's  "  Elements  of  iNIechanism,"  pages  126  to  130. 

69.  The  Klein  Coordinate  Odontograph.  Fig.  -'50  is  de 
signed  to  eliminate  the  labor  of  drawing  pitch  circles  of  large 
radii  by  constructing  the  curve  by  oi-dinatcs  from  a  radial 
line.       The    tables    and    explanation    of    the    method    may    be 


Fig.  30. 


62 


SPECIAL    FORMS    OF    ODONTOIDS    AND    THEIR    LINES    OF    ACTION. 


Fig.  33. 


found  in  Professor  Klein's  "  Elements  of  Machine 
Design,"  page  50. 

70.  Special  Forms  of  Odontoids  and  their  Lines 
of  Action.  Gears  may  be  classified  from  the  forms 
of  rack  teeth,  as  follows : 

System.  Tooth  Curve.         Line  of  Action. 

Involute,      Fig.  31,  A  right  line,  A  right  line. 

Cycloidal,    Fig.  32,  A  cycloid,  A  circular  arc. 

Segmental,  Fig.  33,  A  circular  arc.  Conchoid  of  Xicomedes. 

In  like  manner  other  systems  might  be  derived 
from,  and  classified  by,  the  forms  of  their  rack 
teeth. 

It  is  of  interest  to  note  in  connection  with  the 
first  two  that  any  tooth  of  either  system  ma}^  be 
derived  from  a  right  line.  In  the  cycloidal  system 
the  addendum  of  any  gear '  tooth  will  properl}- 
engage  the  radial  flank  of  some  gear.  If,  there 
fore,  the  addenda  of  any  gear  tooth  be  made  to  fit 
the  dedenda  of  teeth  consisting  of  radial  flanks, 
the  resulting  teeth  must  be  cycloidal.  A  skilled 
mechanic  with  file  and  straight-edge  could  in  this 


CONJUGATE    CURVES. 

manner  produce  the  templet  for  any  de- 
sired cycloidal  tooth  without  the  aid  of 
other  mechanism.  Of  course  such  a 
method  would  i-equire  considerable  skill 
in  producing  a  perfect  tooth,  and  it  is  not  "" 
the  best  means  to  the  end ;  but  it  is  of 
much  interest  to  the  student  as  illustrat- 
ing the  relation  between  the  mechanical 
and  gra[)hic  methods  of  attaining  the  same 
end.  In  like  manner  we  may  produce  templets  for  invo- 
lute teeth  from  the  right  line  rack  tooth  of  the  system. 


03 


71.  Conjugate  Curves.  —  The  curves  of  any  [)air  of 
teeth  being  so  related  as  to  produce  a  uniform  velocity 
ratio  are  called  conjugate,  or  odontoids,  and  if  an}-  tooth 
curve  of  reasonable  form  be  assumed,  a  second  curve 
may  be  obtained  which  shall  be  conjugate  to  the  first. 
By  a  reasonable  form  is  meant  the  conformity  to  the 
foUowdng  principle:  — 

The   normals   to  the  curve  must   come   into  actioi 
consecutively,  as  in  Fig.  o4,  and  not  as  in  Fig.  8"),  in 
which   it   will   be   seen   that   the    normal   E  F   will   pass 
through  the  pitch  point  M,  and  the  point  E  come  into 


Fig.  36. 


64 


WORM    GEARING. 


Eig.  37 


action  before  the  point  C,  which  is  impossible.  Let  C,  Fig.  36, 
be  any  tooth  form  conforming  to  the  above  condition,  and  the 
peripliery  of  disk  A  its  pitch  line.  Suppose  it  is  required  to 
derive  its  conjugate  having  for  its  pitch  circle  the  periphery  of 
disk  B.  This  may  be  obtained  by  a  graphic  process,  as  in 
Art.  28,  page  15,  or  by  the  mechanical  method  known  as  the 
molding  process  of  P'ig.  36.  C  is  a  templet  of  the  given  tooth 
form,  which  is  fastened  to  disk  A ,  and  revolving  in  contact  Avith 
disk  B,  the  disks  maintaining  a  constant  velocity  ratio.  The 
successive  positions  of  C  are  then  traced  on  the  plane  of  disk  B, 
and  the  tangent  curve  will  be  that  of  the  required  conjugate 
tooth. 

The  method  is  applicable  to  all  forms  of  spur  gear  teeth, 
but  to  only  one  form  of  bevel  gear,  the  octoid. 

72.  Worm  Gearing.  A  worm  is  a  screw  designed  to  drive 
a  gear,  called  a  worm  wheel  or  gear,  the  axis  of  the  latter  being 
perpendicular  to  that  of  the  worm.  AiiT.  3,  page  3.  The  sec- 
tion of  a  worm  and  gear  made  by  a  [)lane  perpendicular  to  the 
axis  of  the  gear,  and  including  the  axis  of  the  worm,  is  identical 
witli  that  of  a  rack  and  gear  of  the  same  system  and  pitch.  The 
worm,  or  screw,  may  be  single  threaded,  double  threaded,  etc. 
If  single  threaded,  the  circular  pitch  corresponds  with  the  pitch 


LITERATURE.  65 


of  the  tliread  ;  if  double,  the  circuhir  pitch  will  be  half  the  piteli  of  the  thread,  etc.  To  avoid 
misunderstanding,  it  is  customary  to  speak  of  the  })itch  of  the  thread  as  the  lead. 

A  drawing  of  the  tooth  foi-m  is  required  only  in  sijpcial  cases  of  large  cast  geai-s,  and  the 
usual  representation  is  that  shown  by  Fig.  37. 

The  diameter  of  the  \\()rm  is  commonly  made  equal  to  four  or  five  times  the  circular  pitch, 
and  the  angle  A  varies  from  00°  to  90°. 


Formulas  for  Worm  and  Geai:. 


L   =  Lead  of  worm; 
m  =  Threads  per  inch  in  worm; 
d    =  Outside  diameter  of  worm ; 
d'  =  Pitch  diameter  of  worm; 
W  =  Whole  diameter  of  gear; 
D  =  Throat  diameter  of  gear; 
D'  =  Pitch  diameter  of  gear; 

L  =  —  =  P',  for  single  threads, 
m 

2 
L  =  _  =  2  P',  for  douhle  threads,  etc. ; 
m 

fj  and  r.,  are  dimensions  required  for  the  hob,  or 

cutter,  employed  in  cutting  the  worm  gear; 

C  =  Center  distance; 


p.         -n-  D 
N  +  2' 

O-.lf.-.^; 

o.        N   +  2. 
°  -        P      ' 

d        2. 
h   -  ^        P' 

,     17  . 

r         D  +  d 
^    ~        2 

2. 
P' 

W  =  D  +  2  ( r, 

— 

f  ^1  -  h  cos  -  j 


73.  Literature.  The  following  list  of  books  and  articles  is  published  to  assist  the  student 
who  may  wish  to  jjursue  the  subject  beyond  its  elementary  stage.  Only  those  treatises  have 
been  enumerated  which  are  likely  to  be  accessible  and  useful.     The  great  works  of  Willis, 


66  LITERATURE. 

Rankiiie,  and  Reuleiix  are  omitted,  as  the  student  Avill  derive  more  benefit  from  the  interpreta< 
tion  of  these  worka  by  later  authors  than  by  a  study  of  tlie  original  treatises. 

"The  Mechanics  of  the  Machinery  of  Ti-ansmission,"  revised  by  Professor  Herrmann,  is 
Vol.  III.,  Part  I.,  Sect.  1,  of  Weisbach's  "Mechanics  of  Engineering."  This  work  includes 
one  of  the  most  valualjle  treatises  on  the  subject  of  gearing,  but  it  is  someAvhat  difficult. 
Wiley,  .fS.OO. 

"  Kinematics,"  by  Professor  ^NlacCord,  is  chiefly  devoted  to  the  subject  of  gearing.  It  con- 
tains much  original  matter  of  importance.  No  student  of  the  ;sul)ject  can  afford  to  do  witliout 
this  treatise.     Wiley,  Jif^oOO. 

"Elements  of  Machine  Design,"'  l)y  Professor  Klein,  was  published  for  the  students  of 
Lehigh  University.  Several  chapters  are  devoted  to  gearing,  and  include  some  excellent  tables 
and  problems.  The  Klein  coordinate  odontogi'a[)h  is  fully  illustrated  and  explained.  J.  F. 
Klein,  Bethlehem,  Pa.,  -IG.OO. 

"  A  Treatise  on  Gear  Wheels,"  formerly  called  "  Odontics,"  by  Mr.  Geo.  B.  Grant,  is  one 
of  the  most  valuable  modern  treatises  on  gearing.  It  is  both  theoretical  and  practical.  It  is 
concise,  contains  many  useful  tables,  and  is  well  illustrated.  The  subject  cannot  be  pursued 
to  advantage  without  its  use.     Philadelphia  Gear  Works,  Philadelphia,  $1.00. 

"Practical  Treatise  on  Gearing,"  by  Mr.  O.  J.  Beale.  An  excellent  })ractical  treatment  of 
the  design  and  construction  of  gears.  It  deals  little  with  the  theor}^  but  that  little  is  thor- 
oughly and  simply  taught.     Brown  &  Sharpe  Manufacturing  Company,  Providence,  -fl.OO. 

"Formulas  in  Gearing."  This  is  published  by  the  Brown  &  Sharpe  ^bmufacturing  Com- 
pany, and  contains  many  useful  formulas  for  the  draftsman,  and  valual)le  hints  for  the  cutting 
of  gears.     l|2.00. 


NOTATION    AND    FORMULAS.  67 

"  Elementary  Mechanism, "  l)}-  Professoi-s  Stalil  and  Wood,  is  a  most  comprehensive  text 
book  on  tlie  subject  of  oeaiino'.  It  is  well  classitied,  contains  numerous  examples,  and  is  a 
valualjle  reference  book  for  the  student.      Van  Xostrand,  *2.00. 

"  Gear-cutting  Machinery "  by  Ralph  E.  Flanders  comprises  a  complete  review  of  con- 
tempoi^iry  American  and  European  practice  for  the  forming  and  cutting  of  gear  teeth.  Also 
an  excellent  logical  classification  and  explanation  of  the  principles  involved  in  the  several 
machines  described.     Wiley,  $3.00. 

In  addition  to  the  above  the  student  is  referred  to  the  files  of  the  "  American  Machinist,'' 
"Machinery,"  and  the  "  ^Machinery  Reference  Series,"'  for  many  valuable  articles  relating 
to  the  subject.  These  articles  are  particularly  intei'csting  on  account  of  the  tests  and  novel 
applications  which  they  report. 

The  "Transactions  of  the  American  Society  of  Mechanical  Engineers"  contain  many 
most  valuable  and  ini[)ortant  papers  and  discussions  on  the  latest  theories  and  practice  in 
srearinof. 


74.    Notation  and  Formulas. 

Spur  Gears. 

P'  =  Circular  pitch,  Art.  17,  page  11;  N  =  Number  of  teeth  iu  gear; 

P  =  Diameter  pitch,  Art.  18,  page  11;  n    =  Number  of  teeth  in  pinion; 

D'  =  Pitch  diameter  of  gear;  s    =  Addendum  of  tooth,  Art.  31,  page  17; 

D  =  Whole,  or  addendum,  diameter  of  gear;  f    =  Clearance,  Art.  27,  page  1.); 

d'  =  Pitch  diameter  of  pinion;  t    =  Thickness,  Ai!T.  31,  page  IT: 

d  =  Whole,  or  addendum,  diameter  of  pinion;  p  =  Least  angle  of  pressure,  ARr.  45,  page  33; 


68 

TT  =  3.1416; 

P  =  -Y'    '    D'  ^  P"  ^^'  ^^"^      ' 

N 
P   =  -  ,      P  P'  =  TT,  Akt.  i8,  page  12; 


NOTATION    AND    FORMULAS. 


P^  _     TT      _    1.57 

*    ~   2        2  P  P 


1_ 
P' 


f  =  I  =  ^,  Akt.  31,  page  17; 


Art.  31,  page  IT; 


^,   ,2        PD'+2       N  +  2 
D  =D'+2S  =  D'  +  -  = ^^—  =  — ^— ; 


cos  p  =  ^  / — ;-; — ,  Art.  45,  page  33. 


v'— 


Rg  =  Radius  of  gear, 

Rp  =  Radius  of  pinion, 

Tj    =  Inner  describing  circle, 

r.2    =  Outer  describing  circle, 

rs    =  Intermediate  describing  circle, 

C    =  Center  distance, 


Annular  Gears. 

Rg  =  rs  +  Tj ,  Art.  50,  page  .39; 
Rp  =  r.s  —  r.,,  Art.  50,  page  39; 
C    =  Rg  —  Pp.  Art.  50,  page  39; 
Art.  50,  page  39;   ^^  rnaximum  =  Rg  -  C  ;  r^  minimum  =  C,  Art.  52,  page  40; 

r.,  maximum  =  C  ;  ro  minimum  =  2  C  —  Rg, 

Art.  52,  page  40; 

fg  maximum  =  Rg ;  rs  minimum  =  C,  Art.  51,  page  40. 


Bevel  Gears,  Shafts  at  90°,  Art.  6i,  Page  56. 


A  =  Center  angle  of  pinion; 

B  =  Angle  increment; 

C  =  Angle  decrement; 

E  =  One-half  the  diameter  increment  for  pinion: 

F  =  One-half  the  diameter  increment  for  gear. 


d'         n. 
tan  A   =  -  -  -, 


tan  B 


tan  C 


2  sin  A 


2.25  sin  A. 


E   =  -  cos  A 


F  =  p  sin  A. 


notation  and  formulas.  69 

Bevel  Gears,  Shafts  at  Other  than  90°,  Art.  63,  Page  5G. 

a    =  Angle  of  shafts; 

A    =  Center  angle  of  pinion; 

A'  =  Center  angle  of  gear; 

B    =  Angle  inerenient; 

C   =  Angle  decrement ; 

E    =  One-half  the  diameter  increment  for  pinion; 

E'  =  One-half  the  diameter  increment  for  gear; 

F    =  Dimension  required  for  backing  of  pinion; 

F'  =  Dimension  required  for  backing  of  gear. 

Worm  Gears,  Art.  72,  Pac;e  62. 

L   =  Lead  of  worm ;  ■  1        r,,  ,       .     ,     ,        , 

L   =  —  =  P  for  single  threads; 
m 

m  =  Threads  per  inch  in  worm;  2 

L    =  —  -=  2  P'for  double  thread,  etc., 
m 
d    =  Outside  diameter  of  worm; 

d'  —  Pitch  diameter  of  worm; 

D   =  Thread  diameter  of  gear; 

D'  =  Pitch  diameter  of  gear; 

W  =  Whole  diameter  of  gear. 


N 

—  +  cos  a 

n 

tan  A' 

= 

sin  a 

n 

^  -F  cos  a 

tan  B 

= 

2  sin  A  _  2  sin  A', 
n                   N       ' 

tan  C 

= 

2.25  sin  A  _  2.25  sin  A', 
n                        N         ' 

E 

= 

1a                          C'           1                A' 

p  cos  A                E   =  p  cos  A  ; 

F 

= 

1        ■         A                            C.             1        ■        A' 

sin  A                 F    =  —  sin  A  ; 

°-^P      • 

D^''^^ 

p-=  '° 

N    +   2 

• 

W  =  D  +  2  ^ 

D  +  d 

2 

2. 
P' 

d        2 
h   -  2        p 

• 

•-2   =  ^1  +  3p- 

^) 


70  METHOD  TO  BE  OBSERVED  IN  PERFORMIXG  THE  PROBLEMS. 

CHAPTER  virr. 

PROBLEMS. 

75.  Method  to  be  Observed  in  Performing  the  Problems.  Xo  attempt  should  l)e  made  to 
graphically  solve  the  following  problems  until  the  general  principles  involved  are  ^vell  under- 
stood. 

The  first  requisite  to  this  is  the  mastery  of  Chapter  II.,  on  Odontoidal  Curves ;  and  this 
can  be  best  acquired  1)y  the  drawing  of  the  various  curves,  together  with  a  study  of  their 
characteristics.  Xo  problems  have  been  given  on  this  topic,  ])ut  the  following  couree  of 
study  would  be  desirable :  — ■ 

Having  prescribed  diameters  for  rolling  circles  and  director,  or  pitch  circles,  draw  a  cycloid, 
epicycloid,  and  hypocycloid,  as  described  in  Arts.  5,  G,  and  7,  page  5.  Obtain  a  sufficient 
number  of  points  in  each  case  to  enable  the  curves  to  be  drawn  free-hand  with  considerable 
accuracy,  after  whicli  they  may  l)e  corrected  by  the  use  of  scrolls.  Xext  prescribe  a  i)oint  on 
each  (not  one  already  found),  and  draw  normals  to  each  by  Art.  8,  page  5. 

The  second  method.  Art.  9,  page  5,  is  the  more  practical,  and  should  also  be  studied  by 
drawing  a  small  part  of  each  curve,  beginning  at  a  point  on  the  director  circle. 

It  is  also  desirable  that  one  of  the  epitrochoidal  forms  be  drawn,  and  a  normal  determined. 
Art.  11.  page  6. 

The  problems  are  designed  to  be  solved  on  a  sheet  which  shall  measure  10"  by  14"  within 


PROBLE:\t    1.       CYCLOIDAI.    LIMITING    CASE.  I  1 

tlie  margin  line,  and  the  lay-out  of  these  sheets  is  given  on  Plates  14  and  lo.  Measurements 
are  from  the  margin  line. 

It  is  unnecessary  to  represent  all  the  teeth  in  a  gear,  but  such  as  are  shown  should  he 
drawn  with  the  greatest  accuracy  attainable  by  the  student.  Without  this  care  the  study  will 
avail  one  little,  and  the  time  consumed  in  discovering  errors  will  be  great. 

The  inking  of  the  curves  may  be  omitted  if  time  will  not  admit  of  its  being  well  done; 
but  in  either  case  it  is  desiral)le  to  emphasize  the  curves,  and  distinguish  clearly  between  the 
gears  by  making  a  veri/  light  wash  of  color  on  the  inside  of  the  curve,  the  width  to  be  alx)ut 
one-quarter  of  an  inch.  One  color  may  be  used  for  the  pinion,  and  a  second  for  the  rack  and 
gear. 


Problem  i,  Plate  14, 

Fig.  I. 

Cycloidal  Limiting 

Case. 

Face 

or  Flank  only. 

EXAMPLK. 

D' 

d' 

N 

n 

a                       B 

1 

10 

15 

12 

3  J                 41 

2 

10 

8 

12 

31          n 

3 

121 

7 

21 

3           4 

4 

^•V 

15 

10 

3i                  41 

5 

10 

8 

15 

4|                  4i 

6 

101 

14 

10 

3i                  4i 

7 

6 

24 

12 

03                  4. 

8 

12L 

21 

12 

31                 4.V 

Statement  of  Pkoble.nl  Having  given  the  diameters  of  pitch  circles,  number  of  teeth, 
and  diameter  of  describing  circle,  it  is  required  to  draw  the  teeth  for  pinion,  gear,  and  rack, 
having  arcs  of  contact  equal  to  the  pitch,  and  contact  on  one  side  of  pitch  point  only. 


72  PROBLEM     1.       CYCLOIDAL    LIMITING    CASE. 

Study  Arts.  1  to  26  before  performing  this  problem. 

Operations.  1.  By  Art.  18,  page  11,  determine  the  value  of  N,  n,  D'  or  d'  one  of  which  is 
omitted  from  the  table.      Observe  that  -  =  -. 

d  n 

2.  Draw  center  and  pitch  lines  and  describing  circle.      Lay  "off  the  circular  pitch  on  each 

gear  by  spacing  the  circumferences  into  as  many  parts  as  there  are  teeth. 

3.  Obtain  the  first  point  of  contact  by  laying  off  from  the  pitch  point  on  the  describing 
circle  an  arc  equal  to  the  circular  pitch,  the  direction  being  determined  by  the  rotation 
required.     Art.  16,  page  10.     Art.  21.  page  12.     Arts.  22  and  23,  page  13. 

-1.  AVith  the  above  describing  point,  generate  the  face  and  flank  required.  Arts.  14  and 
15,  page  10. 

5.  Draw  the  working  faces  of  gear  teeth,  and  assuming  the  gear  teeth  to  be  pointed,  draw 
opposite  side  of  each.     Art.  16.  page  10. 

0.  Draw  the  working  flanks  of  the  pinion  teeth,  observing  that  the  depth  must  be  sufficient 
to  ailiait  the  gear  teeth,  but  without  clearance.  Obtain  the  thickness,  and  draw  the  opposite 
sides.     Art.  16,  page  10. 

7.  Draw  the  describing  circle  for  rack.  Obtain  the  first  point  of  contact  between  pinion 
and  rack,  and  describe  the  cycloid  for  rack  teeth.  Construct  rack  teeth.  Art.  25,  page  14. 
Note  that  thickness  of  rack  tooth  must  equal  space  between  pinion  teeth,  or  thickness  of 
gear  teeth,  measured  on  the  })itL'h  line. 

8.  To  determine  points  of  contact  of  conjugate  teeth,  assume  any  point  on  face  of  gear 
tooth,  and  determine,  first,  its  position  when  in  contact  with  the  pinion ;  second,  the  point 
of  the  pinion  tootb  engaging  it.  Since  the  contact  must  take  place  on  the  path  of  contact. 
Art.  21.  page  12,  the  assuniid  point  will  lie  at  the  intei-section  of  this  arc  and  one  described 


PROBLEM    2.       CYCLOIDAL    LIMITING    CASE.  73 

through  the  given  point  from  center  of  gear.  To  solve  the  second,  describe  an  arc  from  the 
center  of  the  pinion  through  the  point  previously  determined,  and  it«  intei-section  with  the 
pinion   Hank   will  be  the  engaging  point  required. 

Next  construct  the  normals  for  each  of  these  points.  Art.  8,  page  5.  They  should  be 
equal  to  each  other,  and  also  to  the  distance  from  the  pitch  point  to  the  point  on  the  path  of 
contact  in  which  they  engage.     Art.  14,  page  10. 

9.  Obtain  the  maximum  angle  of  obliquity,  or  pressure,  between  gear  and  pinion,  pinion 
and  rack.     Art.  24,  page  14. 

Problem  2,  Plate  14,  Fig.  2.  Cycloidal  Limiting  Case.  Face  and  Flank.  Study  Arts. 
26  to  30. 

Statp:ment  of  Problkm.  Tlie  dianietei"s  of  gears,  nunil)er  of  teeth,  and  describing  circles 
being  given,  it  is  re(piired  to  diaw  the  teeth  for  pinion,  gear,  and  rack,  when  the  arc  of 
approach  ==  the  arc  of  recess  =  half  the  circular  pitch,  the  flank  of  gear  being  radial. 

Operations.  1.  Draw  center  lines,  pitch  lines,  and  rolling  circles,  the  second  circle 
being  determined  l)y  Art.  9,  page  6.  Divide  the  pitch  circle  into  the  required  parts  to  obtain 
the  circular  pitch. 

2.  Lay  off  arcs  equal  to  —  on  each  of  the  rolling  circles  to  obtain  the  first  and  last  points 
of  contact,   observing  the  direction  of  rotation  prescribed  in  Fig.  2. 

3.  With  the  [)oint  thus  determined  on  small  rolling  circle,  describe  the  addendum  of  gear 
tooth  and  dedcnduni  of  pinion  tooth.  With  the  ])oiiit  on  the  second  describing  circle  generate 
the  addendum  of  pinion  tooth.  The  dedendum  of  gear  tooth  being  radial  may  then  be  drawn. 
Make  the  dedenda  of  pinion  and  gear  deep  enough  to  admit  the  engaging  addenda,  but  allow 
no  clearance. 


74  PROBLEM  3.   CYCLOIDAL  GEAR. 

4.  Draw  the  working  faces  of  the  phiion  teeth  and  then  the  opposite  faces  to  make  the 
teeth  pointed.  Similarly  draw  the  gear  teeth,  making  them  pointed  also.  The  sum  of  the 
thickness  of  the  teeth  cannot  be  greater  that  the  circular  pitch.  Art.  29,  page  16.  In  this 
case  it  will  be  found  to  be  about  one-hundredth  of  an  inch  less,  which  will  be  the  backlash. 
An  increase  in  the  diameter  of  either  rolling  circle  would  make  the  solution  impossible. 

5.  Draw  the  dedenda  of  pinion  and  gear  teeth. 

6.  The  describing  circles  for  the  rack  teeth  will  l)e  determined  by  Art.  14,  page  10. 
Draw  the  circles  with  their  centers  on  the  line  of  centers,  and  obtain  the  firet  and  last  points 
of  contact.  These  points  should  fall  on  the  addendum  and  dedendum  of  pinion  teeth  already 
drawn,  as  in  Plate  5  at  N  and  0  .  From  these  points  describe  the  addenda  and  dendenda  of 
the  rack  teeth.     The  thickness  of  these  teeth  must  equal  those  of  the  gear. 

7.  Obtain  the  maximum  angle  of  pressure  for  approach  and  recess  between  pinion  and 
gear  and  pinion  and  rack.  It  would  also  be  desirable  to  obtain  the  curve  of  least  clearance 
in  one  case.     Art.  28.  page  15. 

Problem  3,  Plate  14,  Fig.  3.  Cycloidal  Gear.  Practical  Case.  Complete  Chapter  III.  be- 
fore performing  this  problem. 


X  AMPLE 

d' 

N 

n 

a 

A 

c 

1 

9 

18 

12 

4 

5 

31 

2 

8 

21 

12 

4 

4 

^ 

3 

10 

20 

16 

3i 

-4 

3 

4 

8 

22 

12 

3i 

5 

4 

5 

9 

16 

12 

4 

5 

31 

20  12  3  4  41 


PROBLEM  3.   CYCLOIDAL  GEAR.  lO 

Statement  of  Prorlem.  Tlie  diameters  of  pitch  circles  and  rolling  circles  lieing  given, 
and  the  number  of  teeth  known,  it  is  required  to  draw  the  teeth  for  gear,  pinion,  and  i-ack, 
to  obtain  the  maximum  angle  of  obliiiuity,  and  the  arcs  of  approach  and  recess  in  each  case. 
The  teeth  will  be  standard  with  .,\/'  baeklasli.     Art.  81,  page  17.     Art.  71,  page  63. 

Operations.  1.  Figure  the  diameter  of  gear,  circular,  and  diametral  pitch,  Arts.  17 
and  18,  page  11,  and  determine  proportions  of  teeth.     Art.  31,  page  17. 

2.  Draw  center  lines,  pitch  lines,  addendum,  and  dedendum  circles,  and  rolling  circles. 
Divide  the  pitch  circle  into  as  many  parts  as  there  are  teeth,  beginning  to  space  at  the  pitch 
point. 

3.  Beginning  at  the  pitch  point,  describe  j)inion  flank,  gear  face,  gear  flank,  aiul  pinion 
face,  by  Art.  0,  page  5.     See  also  Art.  34,  page  21. 

4.  Lay  off  thickness  of  teeth,  Airr.  31,  page  17,  and  describe  addenda  of  pinion  and  gear 
teeth  by  approximate  method.  Art.  34,  page  22.  Describe  dedenda  l)y  Art.  16,  page  11. 
Draw  fillets.     Art.  31,  page  18. 

5.  Describe  rack  teeth. 

6.  Determine  the  following  for  gear,  pinion,  and  rack  in  tei-nis  of  P'.  Arts.  21  to  24 
inclusive,  pages  12,  13,  and  14,  Art.  32,  page  18. 

Pinion  and  Gear.                      Pinion  and   Rack 
Arc  of  approach 

Arc  of  recess 

Arc  of  contact 

Maximum  angle  of  pressure 


76  PROBLEM    4.       INVOLUTE    LIMITING    CASE. 

Problem  4,  Plate   14,  Fig.  4.     Involute  Limiting  Case.     Study  Arts.  38  to  42. 

StatExMENT  of  Problem.  Number  of  teeth  live  and  six.  Pinion  teeth  pointed.  No 
backhisli  or  clearance.     Arc  of  contact  equal  to  the  circular  pitch. 

This  problem  being  similar  to  tliat  of  Plate  8,  reference  will  be  made  to  that  figure. 

The  case  being  a  limiting  one,  the  distance  between  the  points  of  tangency  of  base  circles 
and  line  of  pressure  must  equal  one-sixth  of  the  circumference  of  the  gear  base  circle,  or  one- 
fifth  of  the  circumference  of  the  pinion  base  circle.     The  tangent  of  the  angle  of  pressure 

will  equal  7-3  =  ^^-^  =  •— = — -7;-::,  but  A  D  =  D  K  C  by  construction,  and  D  K  C  =  tt.     Also  A  F  + 
•^AFDGAF  +  DG  ^ 

D  G  =  5|,  hence,  =  —  =  tan.  of  the  angle  of    pressure.     The  angle  corresponding  to 

this  tangent  is  29°  44'    6".     The   distance   between   the   centers  will  be  ^a~D''^  +  A  F  -f-  D~G^  = 

V^r'-^  +  5:5^  =  6J. 

The  angle  of  pressure  and  distance  between  centers  could  have  been  determined  graphi- 
cally by  laying  off  F  A ,  in  any  direction,  equal  to  the  radius  of  pinion  base  circle,  A  D  perpen- 
dicular to  FA,  and  equal  to  one-fifth  of  pinion  base  circle.  Finally,  D  G  perpendicular  to  A  D , 
and  equal  to  the  radius  of  gear  base  circle. 

Operations.  1.  Draw  the  line  of  centers,  base  circles,  and  line  of  pressure.  Deter- 
mine the  points  of  tangency,  which  limit  the  action  in  either  direction,  and  through  the  pitch 
point,  determined  by  the  intersection  of  the  line  of  centers  and  line  of  pressure,  draw  the 
pitch  circles.  It  is  desirable  now  to  test  A  D  by  proving  it  equal  to  one-fifth  of  the  pinion  base 
circle,  or  one-sixth  of  the  gear  base  circle. 

2.  Draw  the  involute  A  c,  Plate  8,  of  the  gear,  and  D  p  of  tlie  pinion.  Art.  12,  page  7. 
Art.  38,  page  26.  Determine  the  circular  pitch,  and  lay  off  as  many  divisions  as  tliere  are 
teeth  to  be  drawn.     Copy  the  curves  already  drawn. 


PROBLEM  5.   INVOLUTE  PRACTICAL  CASE.  77 

3.  Draw  the  opposite  face  of  pinion  teeth,  making  them  pointed.  To  draw  the  opposite 
faces  of  gear  teeth  proceed  as  follows :  Since  contact  between  the  opposite  faces  must  take 
place  along  the  line  of  action  C  E ,  Plate  8,  the  contact  between  the  engaging  teeth  will  be 
at  E.  At  E  draw  arc  E  1  from  center  G  .  Bisect  this  arc,  and  lay  off  M  and  H  from  this  radial 
bisector  equidistant  with  A  and  C.  Through  these  points  describe  the  curve  of  opposite  face, 
and  draw  the  remaining  tcctli. 

That  portion  of  the  teeth  lying  Avithin  the  base  circle  will  be  radial,  and  extend  sufficiently 
to  admit  the  engaging  teeth,  but  without  clearance. 

4.  Construct  two  rack  teeth.     Art.  40,  page  29. 

5.  Epicj'cloidally  extend  the  gear  teeth  so  as  to  make  them  pointed.  Similarly  extend  the 
rack  teeth,  Init  only  as  nuich  as  the  clearance  for  the  pointed  gear  tooth  \\  ill  pi-rmit.  Aut. 
41,  page  30. 

Problem  5,  Plate  15,  Fig.  i.  Involute  Practical  Cases.  Complete  the  study  of  Chapter  IV. 
■  Statement  of  Pmoblems.  Several  gears  and  racks  are  given  to  describe  involute  teeth 
of  standard  dimensions.  To  determine  the  interference,  if  there  be  any,  and  to  correct  the 
curves  for  the  same. 

Operations.  1.  Draw  three  or  four  teeth  of  gear  A,  and  t\\o  teeth  oi  engaging  pinion 
B,  the  angle  of  pressure  being  15°.  ,  Art.  42,  page  32,  Fig.  16.  Make  contact  at  pitch 
point  in  all  cases.  Correct  for  interference  l)y  epicycloidal  extension.  Art.  31,  page  17. 
Art.  42,  page  30.     Art.  41,  page  32. 

2.  Draw  three  or  four  teetli  of  u'car  A  engasfino'  lack  F  . 

3.  Draw  three  teeth  of  ijininn  B  cno-aoino-  rack  E,  and  correct  rack  teetli  ft)r  interference. 

1  o    o       o  ^ 


78 


PROBLEM    6.       CYCLOIDAL    ANNULAR    GEAR. 


4.  Draw  a  portion  of  gear  C  and  rack  K ,  the  angle  of  pressure  being  20°.     Test  this  for 
interference  by  Art.  45,  page  33,  as  well  as  by  graphic  method. 

5.  Draw  a  few  teeth  of  gear  D,  the  angle  of  pressure  being  15°.     Determine  the  least 
number  of  teeth  that  Mill  engage  it  without  interference. 

Problem  6,  Plate  15,  Fig.  2.     Cycloidal  Annular  Gear.     Study  Arts.  48  to  5Q. 
Example.       D'  d'  N  n  A  a  B 


1 

191 

9 

13 

6 

7 

3i 

H 

2 

19^ 

9 

13 

6 

H 

4 

5h 

3 

191 

9 

13 

6 

6 

H 

5i 

4 

in 

7 

15 

6 

7 

H 

5h 

5 

ITi 

7 

15 

6 

7h 

3 

5i 

Statement  op  Problem.  The  number  of  teeth  and  diametere  of  pitch  and  describing 
circles  being  given,  it  is  required  to  draw  the  tooth  outlines,  and  determine  the  increased  arc 
of  contact  due  to  secondary  action.  The  arc  of  contact,  not  including  that  due  to  the 
secondary  action,  is  equal  to  the  circular  pitch,  and  the  arc  of  approach  equals  the  arc  of 
recess. 

Operations.     1.    Draw  the  center  and  pitch  lines  and  describing  circles. 

2.  Determine  the  circular  pitch,  and  lay  off  half  this  amount  from  the  pitch  point  on  each 
of  the  describing  circles  to  determine  the  first  and  last  points  of  contact. 

3.  Describe  the  curves  of  the  teeth. 


PROBLEM    7.       INVOLUTE    ANNULAR    GEAR.  79 

4.  Determine  the  intermediate  describing  curve,  and  draw  the  same  to  obtain  the  limit  of 
secondary  action. 

5.  Determine  the  maximum  angle  of  pressure  for  approacli  and  recess.     Also  the  angle 
of  pressure  for  the  last  point  of  secondary  action,  and  the  increase  in  the  arc  of  contact. 

Problem  7,  Plate  15,  Fig.  2.     Involute  Annular  Gear.     Complete  Chapter  V. 


Example. 

D' 

d' 

N 

n     An 

gle  of  Pressure. 

B 

1 

15 

'h 

20 

10 

20° 

H 

2 

15 

6 

30 

12 

15° 

7 

3 

IG 

8 

16 

8 

20° 

6 

4 

20 

8 

30 

12 

15° 

7 

5 

24 

18 

24 

18 

20° 

31 

Statement  of  PK()nLE>L  The  pitch  diameters,  number  of  teeth,  and  angle  of  pressure 
being  given,  it  is  required  to  draw  the  tooth  curve,  to  determine  if  tliere  will  be  any  inter- 
ference when  the  addenda  of  pinion  teeth  are  made  standard,  and  finally  the  length  of  the  arc 
of  contact  in  terms  of  P'. 

Operations.     1,    Draw  center  and  pitch  lines,  line  of  pressure,  and  base  circles. 

2.  Make  addenda  of  pinion  standard  if  a  second  engagement  does  not  take  place.  Art. 
56,  page  43,  and  limit  addenda  of  gear  by  Art.  56,  page  43. 

3.  Determiiie  the  arc  of  contact  in  terms  of  P'. 


80 


PROBLEM    8.       CYCLOIDAL    AND    INVOLUTE    BEVEL    GEARS. 


Problem  8,  Plate  15,  Fig.  3.     Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  90°.     Study 
Arts.  57  to  63. 


Example. 

P 

N 

n 

Q 

K 

1 

3 

18 

15 

31 

u 

2 

4 

24 

20 

31 

n 

3 

2 

16 

12 

31 

n 

4 

4 

28 

20 

3 

n 

5 

3 

-21 

15 

3 

n 

6 

2 

14 

12 

31 

n 

7 

3 

21 

18 

H 

u 

8 

2 

18 

14 

4 

n 

9 

4 

20 

16 

3i 

H 

10 

3 

21 

18 

Si 

u 

w 

13 

2 
13 

li 

2 

H 
U 


3 

21 

4' 

2i 
2^ 
31 
3^ 
4" 
21 

3i 


1 
1 
1 


H 

n 
If 
n 
n 


H 


If  involute,  make  angle  of  pressure  15°. 

If  cycloidal,  make  diameter  of  rolling  circles  equal  to  the  elements  of  normal  cone  of 
pinion. 

Statement  of  Peoble^ni.  The  proportions  of  the  gear  l)eing  given  hy  the  tahle,  it  is 
required  to  draw  the  gear  blanks,  describe  the  development  of  the  teeth  on  the  normal  cones, 
and  figure  the  gears. 

Operations,  1.  Having  determined  the  pitch  diameters,  draw  the  gear  blanks.  Art. 
60,  page  48. 


PROBLEM    9.       CYCLOIDAL    AND    INVOLUTE    BEVEL    GEARS.  81 

2.  Describe  two  or  three  teeth  of  each  gear  on  the  developed  surfaces  of  the  outer  and 
inner  normal  cones.     Art.  60,  page  48. 

3.  Figure  the  geai-s,  Art.  61,  i)age  51. 

Problem  9,  Plate  15,  Fig.  4.     Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  other  than 
90°.     Study  Art.  63. 


EXAMPI.K 

a 

P 

N 

n 

Q 

J 

K 

L 

M 

H 

w 

X 

U 

V 

Y 

1 

40° 

3 

24 

15 

9 

SI 

2i 

i 

n 

g 

lA 

3 

i 

2i 

li 

2 

45° 

3 

24 

15 

9 

'^i 

2i 

i 

If 

§ 

n 

3 

i 

2 

u 

3 

50° 

4 

34 

24 

81 

8 

2 

i 

n 

i 

n 

0 

i 

n 

u 

4 

55° 

3 

27 

21 

9 

8 

21 

3 

n 

h 

n 

3i 

i 

li 

li 

5 

60° 

2 

20 

12 

8i 

7i 

2.1; 

i 

n 

1 

2 

3i 

i 

2i 

li 

If  involute,  make  angle  of  pressure  15°. 

If  cycloidal,  make  diameter  of  rolling  circles  equal  to  the  elements  of  normal  cone  of 
pinion. 

Statement  of  Problem.  The  proportions  of  the  gear  being  given  by  the  table,  it  is 
required  to  draw  tlie  gear  blanks,  describe  tlie  teeth  on  the  development  of  the  normal  cones, 
and  figure  the  gear. 

Operations.  1.  Determine  the  pitch  diameters  from  above  table,  and  draw  the  gear 
blanks. 

2.  Describe  two  or  three  teeth  of  each  gear  on  the  developed  surfaces  of  the  outer  and 
inner  normal  cones. 

3.  Fiourc  the  geai-s.  _,,„     U-RA'^Y 

8   AT:  T'/CFERSC^L-EOE 
SA.TA  LAnD.;HA.  CALIFORNIA 

/•.(^.?...7-0. 


IIS^DEX. 


Keferences  ar;  to  pages. 


Addendum  dpfined,  12;  proportion  for,  17. 

Anjjle  decrement,  52. 

Angle  increment,  52. 

Angle  of  edge,  51 ;  of  face,  51. 

Angle  of  obliquity,  or  pressure,  14;  affected  by  rolling  circle, 
18;  con.stant,  28;  for  involute,  31;  influence  of,  .'{.3; 
method  for  determining,  3;},  reduced  in  annular  gear- 
ing, 40. 

Annular  gear,  notation,  and  formulas,  08;  epicycloidal  prob- 
lem, 78;  involute  problem,  79. 

Annular  gearing,  38;  secondary  action  in,  38;  interchangeable 
with  spur  gearing,  42  ;  involute  system  of,  43. 

Approaching  action  detrimental,  17. 

Approximate  cycloidal  curves,  22. 

Approximation,  Tredgold,  47;  by  circular  arcs,  22. 

Arc  of  approach  defined,  13. 

Arc  of  contacrt  defined,  13;  relation  to  circular  pitch,  16. 

Arc  of  recess  defined,  13. 

Backing,  52. 

Back  cone,  48. 

Backlash  defined,  !(!;  dimensions  for,  18. 

Base  circle  defined,  7,  27. 

Base  of  system,  21 ;  in  annular  gearing,  42. 


Beale's  "  Practical  Treatise  on  Gearing,"  66. 

Bevel  gear  defined,  2;  Theory  of,  45;  character  of  curves 
employed,  4^!;  drafting  the,  48;  blank,  49;  length  of 
face,  49;  figuring  the,  51;  table  for,  53,  54,  55;  chart  for 
plotting  curves,  (>7 ;  notation  and  formulas,  68;  problems, 
80,  81. 

Bevel  gears  with  axes  at  any  angle,  56. 

Bilgram,  Hugo,  inventor  of  octoid  tooth,  47;  machine  for  cut- 
ting bevel  gear  teeth,  47,  67 ;  exhibit,  67. 

Brown  &  Sharpe  publications,  (>6. 

Circular  pitch  defined,  11. 

Character  of  curves  in  bevel  gearing,  46. 

Clearance  defined,  15;  proportion  for,  17. 

Clock  gears,  17. 

Conchoid  of  Nicomedes,  62. 

Conditions  governing  the  practical  case,  16. 

Conjugate  curves  defined,  9,  63. 

Constant  angle  of  pressure,  28. 

Constant  velocity  ratio  defined,  1. 

Conventional  representation  of  spur  gears,  25. 

Contact,  i)oint  of,  5;  radius,  5;  path  of,  12;  arc  of,  13. 

Coiirdinate  odontograph,  61. 

Crown  gear,  47. 


8a 


84 


INDEX. 


Curtate  epitrochoid,  6. 
Curve  of  least  clearance,  15. 
Curves,  odoutoidal,  4. 
Cutting  bevel  gear  teeth,  67. 
Cutting  angle,  51. 

Cycloid  defined,  4 ;  problem  relating  to,  70. 
Cycloidal  action,  Theory  of,  8. 

Cycloidal  curves,  second  method  for  describing,  5;  approxi- 
mated, 22. 
Cycloidal  system  of  annular  gearing,  38. 
Cycloidal  annular  gear  problem,  78. 
Cycloidal  bevel  gear  problem,  80,  81. 
Cycloidal  limiting  case  problems,  71,  73. 
Cycloidal  practical  case  problem,  74. 

Dedendum  defined,  12 ;  proportions  for,  17. 

Defects  of  involute  system,  35. 

Describing  circle  defined,  4;  a  path  of  contact,  12;  maximum 
and  minimum,  16;  inHuenee  on  shape  and  efficiency  of 
teeth,  18;  relation  to  interchangeable  gears,  20. 

Describing  disk,  8. 

Describing  point,  4. 

Describing  cone,  45. 

Describing  cylinder,  45. 

Describing  i-adius,  5. 

Description  of  bevel  gear  table,  53. 

Developed  pitch  circle,  50. 

Development  of  normal  cone,  40. 

Diameter  pitch,  11. 

Director  circle,  5. 

Double  contact  in  annular  gearing,  .39. 

Double  generation  of  epicycloid  and  hypocycloid,  G. 

Drafting  bevel  gears,  48. 


"Elements  of  Machine  Design,"  66. 

"Elementary  Mechanism,"  67. 

Epicycloid  defined,  5;  second  method  for  describing,  5; 
double  generation,  6;  spherical,  45;  problem  relating 
to,  70. 

Epicycloidal  extension,  .30. 

Epitrochoid  defined,  0;  curtate,  6;  prolate,  7;  problem  relat- 
ing to,  70. 

Exterior  (outer)  describing  circle,  39;  limitations  of,  40,  41. 

Face  gearing,  2. 

Face  of  gear,  24. 

Face  of  tooth,  12. 

Flank  of  tooth,  12;  radial,  18. 

Figuring  bevel  gears,  51. 

Fillet,  18;  size  of,  18. 

"  Formulas  in  Gearing,"  66. 

Formulas  for  worm  and  gear,  05. 

Formulas,  Notation  and,  67. 

Gearing,  1. 

Gear  arm  iiroportions,  67. 

Gears,  interchangeable,  20 ;  face  of,  24 ;  comparison  of,  24. 
Generating  point,  4. 
Generating  radius,  5. 

Grant,  Geo.  B.,  bevel  gear  chart,  53 ;  three  point  odontograph 
58;  involute  odoutograiih,  59;  "  Odontics,"  66. 

Hyperboloid  of  revolution,  2. 

Hyperbolic  gears,  2. 

Hypocycloid  defined,  5;  second  method  for  describing,  5;  a 

radial  line,  6;  double  generation,  6;  spherical,  45;  pi'ob- 

lem  relating  to,  70. 


INDEX. 


86 


Influence  of  tlie  angle  of  pressure,  33. 

Influence  of  the  diameter  of  rolling  circle  on  shape  and  effi- 
ciency of  teeth,  18. 

Inner  describing  circle,  39;  limitations  of,  40,  41. 

Inner  normal  cone,  48,  50. 

Instantaneous  radius,  4. 

Intermediate  describing  circle,  :i'.>,  limitations  of,  40. 

Internal  gear,  see  annular  gear. 

Interference,  .32 ;  in  annular  gearing,  43. 

Interchangeable  gears,  20. 

Involute,  4;  defined,?;  system,  20;  curves,  character  of,  27: 
rack,  28;  system  of  annular  gearing,  43;  annular  gear 
problem,  7!);  bevel  gear  tooth,  4(;;  bevel  gear  problems, 
80,81;  limiting  case,  20;  limiting  case  problem,  7(5;  prac- 
tical case,  .'50;  practical  case  problem,  77. 

Involute  action,  Tlieory  of,  2(5;  limit  of,  28. 

Involute  gearing,  defects  of  system,  35. 

Involute  teeth,  epicycloidal  extension  of,  30. 


"Kinematics,"  MacCord's,  (5(5. 
Klein's  cotlrdinate  odontograph,  (51; 
Design,"  06. 


"  Elements  of  Machine 


Law  of  tooth  contact,  10. 

Lead  of  screw,  (5(5. 

Least  angle  of  pressure,  method  for  determining,  33. 

Least  number  of  teeth  in  annular  gears,  42. 

Limit  of  involute  action,  28. 

Limiting  case,  cydoidal,  10,  14;  involute,  2!);  annular  gear- 
ing, 38. 

Limitations  of  intermediate,  exterior,  and  interior  describing 
circle,  40,  41. 

Line  of  action  a  great  circle,  47. 


Literature,  65. 
Logarithmic  spiral,  01. 

MacCord's  "  Kinematics,"  66. 

Method  for  determining  least  angle  of  pressure,  33. 

Method  to  be  observed  in  jierforming  problems,  70. 

"  Mechanics  of  Engineering,"  (5(5. 

"  Mechanics  of  the  Machinery  of  'I'lansmission,"  (50. 

Normal  defined,  4;  to  construct,  5;  law  governing,  03. 
Normal  cone,  48  :  development  of,  49. 
Notation  and  formulas,  (57. 

Obliijuity,  angle  of,  14. 
Octoid  bevel  tooth,  47,  04. 

"  Odontics,"  "  A  Treatise  on  Gear  Wheels,"  Grant's,  (50. 
Odontoid  defined,  1 ;  special  forms  of,  (52. 
Odontoidal  curves,  4:  problems  relating  to,  70. 
Odontographs  and  odontograph  tables,  .57. 
Odoiitograi)h,  Willis,  00;    Grant  involute,  59;   Grant  Three- 
point,  .58;  Robinson,  (il ;  Klein,  01 ;  coordinate,  61. 
Outer  describing  circle,  39:  limitations  of,  40. 
Outer  normal  cone,  48. 

Path  of  contact  defined,  12;  affected  by  rolling  circle,  18;  a 

right  line.  28. 
Path  of  approach  defined,  15. 
Path  of  recess  defined,  15. 
Pitch  cone,  48. 
Pitch  line,  10. 
Pitch  point,  9,  10,  13,  27. 
Pitch  circular,  11 ;  diameter,  11. 
Planed  bevel  gear  teeth,  07. 


86 


INDEX. 


Positive  rotation  defined,  11. 

Practical  case,  conditions  governing  the,  16;   cycloidal,  21; 

involute,  30;  annular,  42. 
"  Practical  Treatise  on  Gearing,"  G6. 
Pressure,  angle  of,  14. 
Prolate  epitrochoid,  7. 
Proportions  for  standard  tooth,  17. 
Problems,  method  to  be  observed  in  performing,  70. 

Rack,  14;  involute,  28;  gears  classified  by,  62. 

Radial  flank,  18;  as  base  of  system,  21,  62. 

Radius,  describing,  5;  ccntact,  5. 

Rankine,  66. 

Reuleux,  66. 

Robinson  odontograph,  61. 

Rolling  circle,  see  describing  circle. 

Rotation,  positive,  1, 

Screw  gearing  defined,  3. 

Scroll,  use  of,  11. 

Second  method  for  describing  cycloidal  curves,  5. 

Secondary  action  in  annular  gearing,  38,  41. 

Seo;mental  system,  62. 

Skew  gear  defined,  3. 

Spiral  gear  defined,  3. 

Special  forms  of  odontoids,  62. 


Spherical  epicycloid,  45. 

Spherical  hypocycloid,  45. 

Spur  gear  defined,  2;  illustrated,  10;  having  action  on  one 
side  of  pitch  point,  10;  having  action  on  both  sides  of 
pitch  point,  14;  conventional  representation,  25;  inter- 
changeable with  annular  gears,  42 ;  notation  and  formu- 
las, 67. 

Theory  of  cycloidal  action,  8. 
Theory  of  involute  action,  26. 
Thickness  of  tooth,  17. 
Three-point  odontograph,  58. 
Tooth  contact,  law  of,  10. 
To  construct  a  normal,  5. 
Tredgold  approximation,  47. 

Unsymmetrical  teeth,  37. 
Use  of  bevel  gear  table,  53. 

Velocity  ratio  constant,  1 :  not  affected  by  increase  of  center 
distance  in  involute,  28. 

Weisbach's  "  Mechanics,"  66. 

Willis,  odontograph  of,  60;  writings  of,  66. 

Worm  gearing  defined,  3,  64;  notation  and  formulas  for,  69. 

Worm  wheel,  64. 


Plate  I. 

Cycloid,  Epicycloid,  Hypocycloid  and  Involute  curves. 

REFERENCES    TO    TEXT. 

Art.  4,  Page  4.  Art.  8,  Page  5. 

5,  4.  9,  5. 

6,  5.  12,  7. 

7,  5. 


Plate  l. 


Fig. 3 


Plate   2 


Plate  2. 

Epitrochoidal  curves.     Double  generation  of  Epicycloid  and 
Hypocycloid.     Approximate  method. 

EEFEREXCES    TO    TEXT. 

Abt.  10,  Page     6. 
11,  6. 

34,  22. 


-2l?l-E    FOR     EPltMOi'i'S 


Plate  3. 

Mechanical  method  for  describing  Odontoidal  curves. 

REFERENCES    TO    TEXT. 

Art.  13,  Page     8. 
15,  10. 

21,  12. 


Plate  3, 


Plate  4. 


Plate  4. 


Cycloidal  Gear,  Pinion  and  Rack  having  action  on  one  side  of 
pitch  point.     Limiting  case. 


TSF.li'F.KENCES  To    TEXT. 

i'a.-f  in.  Ar.T.  24, 

10.  2.-,, 

18,  1:;.  -M, 

19,  1:;.  3ij, 


14. 

14. 
14. 
•24. 
•37. 


Plate  5. 


Cycloidal  Gear,  Pinion  and  Rack  having  action  on  both  sides 
of  the  pitch  point.     Limiting  case. 


KEFERESCES   TO   TEXT. 


Art.  23,  Page  13. 

26,  14. 

28,  15. 

32,  19. 


Art.     36,  Page  24. 

411,  38. 

Pkob.     2,  74. 


Plate   5. 

u!   r 

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Plate   6. 


Plate  6. 

Cycloidal  Gear,  Pinion  and  Rack.     Practical  case. 

HKKKIIKNCKS    TO   TE.XT. 

AiiT.  34,  Page  21. 
36,  24. 


Plate  7. 

Involute  Gear  and  Pinion.     Limiting  case.     Mechanical  method 
for  describing  the  Involute. 

REFERENCES    TO    TEXT. 

Art.  38,  Page  26. 
39,  27. 

42.  31. 


Plate  7. 


TI 


f 


J 


PLATE    8. 


Plate  8. 

Involute  Gear,  Pinion  and  Rack.     Limiting  case. 

EEFKKENCES    T(J    TEXT. 


Art.  3fl,  Page  28. 
40,  29. 


Art.    41,  Page  30. 
Peob.     4,  76. 


Plate  9. 

One  Pitch  Involute  Gear  and  Pinion,  showing  Interference. 

REFERENCES    TO    TEXT. 

Art.  42,  Page  30.  Art.  44,  Page  33. 

43,  32.  46,  35. 


Plate  9. 


GEAR   30  TEETH 


1   PITCH  INVOLUTE  GEAR  &  PINION 
SHOWING  INTERFERENCE 


Plate  lo. 


Plate  10. 

One  Pitch  Involute  Pinion  and  Rack,  showing  Interference. 

IlEFEKENCES    TO    TEXT. 


Art.  42,    Page  30. 
43,  32 


AuT.  44,   Puge  33. 
46,  35. 


1   PITCH  INVOLUTE  PINION  i  RACK 
SHOWING  INTERFERENCE 


Plate  II. 

Annular  Gearing. 

REFERENCES    TO    TEXT. 

Art.  49,  Page  38.  Art.  52,  Page  41. 

50,  39.  54,  42. 


Plate  11 


Plate  12. 

Annular  Gearing.     Special  cases. 

KEFERENCES    TO   TEXT. 

Art.  50,  Page  39. 

51,  40. 

52,  40. 


Plate  12 


x.Yv 


Plate  13. 

Bevel  Gearing. 

KEFERENCES    TO    TEXT. 

Art.  59,   Page  47.  Art.  60,  Page  48. 


Plate  13 


NKS 


i 

■Fig.  2 


^  / 


Fig.  4 


Plate  14 


Plate  14. 

Problems  i  to  4  inclusive. 

REFERENCES   TO    TEXT. 


Art.     75,  Page  70. 

Prob.     1,  U. 

2,  73. 


Prob.  3,  Page   74. 
4,  76. 


I  ^-i-rc^..^' 


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Plate  15. 

Problems  5  to  9  inclusive. 

REFERENCES    TO    TEXT. 

Art.     75,  Page  70.  Prob.  7,  Page  79. 

Prob.     5,  77.  8,  80. 

6,  78.  9,  81. 


Plate  15 


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